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The total reflection of a sound pulse of arbitrary form
WAVE MOTION 2 (1980) 247-253 0 NORTH-HOLLAND PUBLISHING
THE TOTAL
COMPANY
REFLECTION
OF A SOUND PULSE OF ARBITRARY
FORM
L1.G. CHAMBERS School of Mathematics and Computer Science, University College of North Wales, Bangor, Gwynedd, United Kingdom
Received 17 July 1979, Revised 4 February 1980
The problem of a sound pulse of arbitrary form incident on a half space with an angle of incidence greater than the critical angle is discussed. Formulae are obtained for the transmitted and reflected pressure fields. An expression is obtained for the energy flux across the interface, and it is shown that the net energy flow per unit area over all time is zero.
1. Introduction
The theory of what happens when a plane harmonic sound wave is incident upon the plane interface between two semi-infinite homogeneous isotropic media is well known [6]. A similar plane harmonic sound wave of diminished amplitude is set up as a specular reflection, and a plane harmonic sound wave is transmitted into the reflecting medium, the direction of which obeys Snell’s Law. When the velocity of sound in the medium in which the sound wave is incident is, however, less than the velocity of sound in the second medium and the angle of incidence exceeds a certain critical angle, which depends upon the ratio of these two velocities, then the sound wave in the reflecting medium is evanescent, being attenuated away from the interface, and the phenomenon termed total internal reflection occurs. This phenomenon would arise, for example, when a sound wave is incident from air to water. There does not appear to be any treatment in the textbooks of acoustics (12 have been inspected) of what happens when a plane sound pulse of arbitrary form is incident from a direction which would be associated with total internal reflection in the case of a harmonic plane wave. A similar problem involving the reflection of waves at the interface between two elastic solids has been considered by Friedlander [4]. In [4], the results are obtained by the solution of integral equations for a plane harmonic function using the half plane analogue of Poisson’s formula, after matching the fields on both sides of the interface. It is possible, by means of the Fourier integral to derive the reflected and transmitted pressure fields associated with an incident pressure pulse of arbitrary form, as explicit expressions, but rather surprisingly this does not seem to have been done. The present work follows on from that of Arons and Yennie [l-3].
2. Formulation
of problem
Exactly the same formulation and notation as that of Arons and Yennie [l] will be used. The half space y > 0 is filled with a medium of density p1 and sound velocity Cr and the half space y < 0 is filled with a medium density p2 and sound velocity C2 (>C,). An incident harmonic pressure field
(1) 247
L1.G. Chambers / Total reflection of a sound pulse
248
gives rise, in the case of total internal reflection to a reflected harmonic pressure field of the form exp[io(t-clix and a transmitted
-Piy)+i2EIWI/W],
(2)
harmonic pressure field
2 cos E exp[iw(t-a!lx)+i&IWI/W
+lolP2,yl,
(3)
where 13~is the angle of incidence, and (Y~= CT1 sin f%, pzl = (sir? &/C:
p1= c,-’ cos 81, - l/C$)“*,
E=
tan-1(P2dP~p2).
Suppose now that the incident pressure pulse is defined by PI=fo-%x+PlY),
where f(t)=
O”4(w) era’dw.
(4)
I -al
It will be convenient to write r=t--alX.
(5)
The reflected pressure field will then be given by PR=
~_~~(W)exp[iw(r-P1y)+i2~!$]dw
(6)
00 = cos 2E
4(w) exp{iw(r-piy)}dw I --03
+isin 2E
fm
lw14(u)exp{iw(7-j3ry)}do. -cc w
(7)
Now, following Arons and Yennie, the second integral on the right-hand side of (7), may be simplified by the use of the Dirichlet discontinuous factor:
bl w=
(Ti)-lp
dt, I eiwi 5 *
(f-3)
--co
where the symbol P stands for ‘principal value’. It follows that PR=COS2Ef(f-(YlX-ply)+
sin 2.5 -P rr
4(w) eiw(r--P1y+t)dm.
(9)
The second expression on the right-hand side of (9) can, however be simplified further, to yield PR = cos 2&f(t - (Y~X-P1y)+sin
2~ g(t-alx
--ply),
(10)
where g(u)=iP
Icof(u+4’)d5 5 . -m
(11)
L1.G. Chambers / Total reflection of a sound pulse
249
The transmitted pressure field will be given by m
101
I
pT=2cOs&
iWr+ie-+]til&,y w
--a3
I
do.
(12)
From the convolution theorem for Fourier Integrals, it follows that 00 P==2COSE
I
_m f(7
(13)
- SM (5) dS,
where (14)
=eieI
0
co
--co
=
exp(w(i5 +&y )} do + e-”
e”(-i[
-
pZly
)-’ + e-“(it
- &,y
I
exp{w (il
--oo
)-l
-
PZ,Y )I dw.
(15)
-1 &,y cos E + 5 sin E
(16)
C-n
Thus the transmitted pressure field is given by m
-2lr-i
I
cos & _J(l-W5)
P2,y cos E + 5 sin E dS. I
(P2rY I2 + c2
I
(17)
3. Fluid particle velocity For an acoustic disturbance, the pressure and the fluid particle velocity V, are related through a velocity potential 4, by the relations [5] v=vq5, p = -p
(18)
aq5/dt
(19)
av/at.
(20)
giving the result Vp = -p
This is of course the linearized equation of motion of the fluid. The two components of velocity will obey the relations
ap/ax= -paV,/dt
(204
aplay= -pW,/dt.
Gob)
and
L1.G. Chambers / Total reflection of a sound pulse
250
Consider first the disturbance in the half space y > 0. The total pressure field is given by (21)
~~=f(t-culx+P~y)+cos2ef(t-cw~x-Ply)+sin2~g(t-(~1X-B1y). It follows without any difficulty that vx
=
(~lIPl)Pl
(22)
and that
V,=~P~lp~~[co~~~f~t-(~~~-P~~~+s~~~~g~t-~~~-P~y~-f~~-~~x+P~y~l. In the half space y < 0, the pressure associated with the transmitted PT = -26’
field is given by
f0 -
cos &
(23)
(24)
It follows without any difficulty that vx
=
(25)
bl/P2)PT.
The calculation of V, is somewhat more complicated.
av, 1 -_=--at
p2
We have
aP
ay
=-
(26)
Now
a G [
P2,y cos F + [ sin F cos E k’- @2rY)21-2P2ry sin e I =P2r KP2rY)2+~212 @2rY12+C2 a
p2ry sin
E - 5 cos F I (P2,y)2+52
=%J Thus
?.5 2&!.E np2
co
I
f(t-cr1x-O-
--oD
a
1
&y sin F -C cos 8 dl.
a6 [
032rY)2+L2
Integrate by parts, and it follows that
7$= 2p2,z2 w2
because, provided
m
I
f'(t--1x
--oo
-5)
1
P2,y sin E - 5 cos E d5,
[
032rY)2+52
f is always finite, the integrated part vanishes at the limits. Thus (27)
251
LLG. Chambers / Total reflection of a sound pulse
4. Energy flow
The energy flow E for an acoustic disturbance is given by the equation E =pV.
(28)
The boundary conditions at an interface, which were in fact used in the derivation of the expressions (2) and (3), are that both the pressure, and the normal component of velocity, are continuous across the interface. It follows therefore that the normal component of energy flux across the interface will be continuous. This quantity is given by pV,. As the expressions for p and V, in y > 0 are slightly simpler, it will be easier to calculate the energy flux across the interface from these. For y>O, E,=[f(t-cvlx+~ly)+cos2~f(t-~1~-~ly)+sin2~g(t-culx-~ly)] (2%
x(Pl/pl)[Cos2Ef(f-(Y1X-PlY)+Sin2Eg(f-CYlX-P1Y)-f(t-(Y1X+PlY)l.
The flux across the interface is, on putting y = 0, therefore given by (&/pi)[f{l
+cos 2E}+ g sin 2e][f(cos 2~ - 1) + g sin 283 =
= (P1lP&?
-f’) sin22e +fg sin 4~).
(30)
The net energy flux over all time across the interface at a particular point is therefore {[go - a1x)]2-[f(t-~1~)]2}dt+sin4~
Now it is a well-known result that [2] 03 _m {f(t) +ig(t)}’ dt = 0. I
dt
. I
(31)
(32)
Thus it follows that the net energy flux over all time at a particular point is zero. In exactly the same way, it follows by integrating with respect to x that the net energy flux across the surface at a particular time is zero.
5. Some examples
(a) If f(t) = cos wt, we have (33) It follows from this that an incident pressure wave given by cos w(t--six
+&y)
(34)
will give rise to a reflected pressure wave of the form cos[w(r-aix-ply)+2&].
(35)
Expressions (34) and (35) are in fact the real parts of the expressions (1) and (2) respectively. Similarly, it may be shown that the correct expression is obtained for the pressure of the transmitted disturbance. For the
L1.G. Chambers / Total reflection of a sound pulse
252
harmonic field, the energy flux across the interface is given by (p1/p>[{cos2 w7 -sir? wr} sin2 2s - cos 07 sin w7 sin 4~1.
(36)
It can easily be verified that the mean value of this over a time (2n/w) or over a strip xo
(37)
and formally (38) This step can in fact be made legitimate by taking s(t) as the limit of some sequence. The reflected pressure field will be pn=S(t-(~i~-~iy)cos2~-sin2~/{t-~~~-/3~y}.
(39)
It follows from (17), that the associated transmitted pn= -26’
cos & (&ry cos &+ (t -six)
pressure field is given by
sin .5)/{p2,y)2 + (t-(Y1X)2}.
(40)
Inspection of (39) and (40) reveals that the reflected and transmitted fluid are ahead of the incident pulse on the interface, and the transmitted and reflected field exist for all time, even when the incident pulse has a well defined front. Although this phenomenon is apparently rather suprising, the reason for it is quite straightforward. The velocity of propagation of the incident pulse, parallel to the interface is C1 sin &, whereas the velocity of propagation in the reflecting medium is C2. Thus the transmitted disturbance in the second medium will run along the interface ahead of the incident disturbance and will therefore cause a reflected disturbance which will also run ahead of the incident pulse. This is because of the condition for total internal reflection C2 > Ci cosec 13~.
(41)
f(t) = H(t) cos(wt + y),
(42)
(c) If
where H(t) is the Heaviside step function, we have g(u)=$P
O”H(u +~)cos(ou +w<+y) I -cc
= 1 cos(wu + y) P 7r
r
dl Z-P1 m cos(wu + or + y) dl IT I --oo s (wu +y) P
(44)
Now
P
(43)
ifu>O_
L1.G. Chambers / Total rejection of a sound puke
Alternatively P
253
if u < 0, the integrand is non singular over the range of integration and so it is possible to write *
J--u
6,ydl=
ydl=
-Cilouj.
Also P
msin w 5 dl = -si(-wu). I-I4 5
Thus g(u) -- f sin(wu + v)si(-wu)-i
(45)
cos(wu + y)Cilwu).
Similar comments to those given after (40) apply here. It will be noted that there will be a singularity in g(u) at u =O. The expression (17) for the transmitted pressure field can also be evaluated but the process is rather long and complicated and will be omitted here. (d) If f(t) = H(t)
-H(t - r*), r* > 0,
we have
(46) In exactly the same way, the transmitted pressure field is given by -26’
=
cos E
7*-1p2,y cos t + 5 sin I --T &rY12+t2
-27r-’ cos 8
Cf
t dy =
‘*PT~os tan-l (&)
= TT-’ cos2 E[tai’fg)
-tan-l(&)]
+ d sin logk2 + @2ry)zl]
+r-lCOS
E
sin 8 log[ (7 -‘::,!~$~,,2].
(47)
References [1] A.B. Arons and D.R. Yennie, “Phase distortion of acoustic pulses obliquely reflected from a medium of higher sound velocity”, _T.Acoust. Sot. Amer. 22, 231-237 (1950). [2] P.L. Butzer and R.J. Messel, Fourier Analysis and Approximation, Vol. 1: One Dimensional Theory, Birkhauser, Verlag Base1 (1971) 316. [3] W.M. Ewing, W.S. Jardetsky and F. Press, Elastic Waves in Layered Media, McGraw-Hill, New York (1957) 90. (1945). [4] F.G. Friedlander, “On the total reflection of plane waves”, Quart. .I. Mech. Appl. Math. 1,376-383 [SJ L.D. Landau and E.M. Lifschitz, Fluid Mechanics, Pergamon, London (1959) 246. [6] J.W.S. Rayleigh, The Theory of Sound Vol. 2, Dover, New York (1945) 83.
THE TOTAL
COMPANY
REFLECTION
OF A SOUND PULSE OF ARBITRARY
FORM
L1.G. CHAMBERS School of Mathematics and Computer Science, University College of North Wales, Bangor, Gwynedd, United Kingdom
Received 17 July 1979, Revised 4 February 1980
The problem of a sound pulse of arbitrary form incident on a half space with an angle of incidence greater than the critical angle is discussed. Formulae are obtained for the transmitted and reflected pressure fields. An expression is obtained for the energy flux across the interface, and it is shown that the net energy flow per unit area over all time is zero.
1. Introduction
The theory of what happens when a plane harmonic sound wave is incident upon the plane interface between two semi-infinite homogeneous isotropic media is well known [6]. A similar plane harmonic sound wave of diminished amplitude is set up as a specular reflection, and a plane harmonic sound wave is transmitted into the reflecting medium, the direction of which obeys Snell’s Law. When the velocity of sound in the medium in which the sound wave is incident is, however, less than the velocity of sound in the second medium and the angle of incidence exceeds a certain critical angle, which depends upon the ratio of these two velocities, then the sound wave in the reflecting medium is evanescent, being attenuated away from the interface, and the phenomenon termed total internal reflection occurs. This phenomenon would arise, for example, when a sound wave is incident from air to water. There does not appear to be any treatment in the textbooks of acoustics (12 have been inspected) of what happens when a plane sound pulse of arbitrary form is incident from a direction which would be associated with total internal reflection in the case of a harmonic plane wave. A similar problem involving the reflection of waves at the interface between two elastic solids has been considered by Friedlander [4]. In [4], the results are obtained by the solution of integral equations for a plane harmonic function using the half plane analogue of Poisson’s formula, after matching the fields on both sides of the interface. It is possible, by means of the Fourier integral to derive the reflected and transmitted pressure fields associated with an incident pressure pulse of arbitrary form, as explicit expressions, but rather surprisingly this does not seem to have been done. The present work follows on from that of Arons and Yennie [l-3].
2. Formulation
of problem
Exactly the same formulation and notation as that of Arons and Yennie [l] will be used. The half space y > 0 is filled with a medium of density p1 and sound velocity Cr and the half space y < 0 is filled with a medium density p2 and sound velocity C2 (>C,). An incident harmonic pressure field
(1) 247
L1.G. Chambers / Total reflection of a sound pulse
248
gives rise, in the case of total internal reflection to a reflected harmonic pressure field of the form exp[io(t-clix and a transmitted
-Piy)+i2EIWI/W],
(2)
harmonic pressure field
2 cos E exp[iw(t-a!lx)+i&IWI/W
+lolP2,yl,
(3)
where 13~is the angle of incidence, and (Y~= CT1 sin f%, pzl = (sir? &/C:
p1= c,-’ cos 81, - l/C$)“*,
E=
tan-1(P2dP~p2).
Suppose now that the incident pressure pulse is defined by PI=fo-%x+PlY),
where f(t)=
O”4(w) era’dw.
(4)
I -al
It will be convenient to write r=t--alX.
(5)
The reflected pressure field will then be given by PR=
~_~~(W)exp[iw(r-P1y)+i2~!$]dw
(6)
00 = cos 2E
4(w) exp{iw(r-piy)}dw I --03
+isin 2E
fm
lw14(u)exp{iw(7-j3ry)}do. -cc w
(7)
Now, following Arons and Yennie, the second integral on the right-hand side of (7), may be simplified by the use of the Dirichlet discontinuous factor:
bl w=
(Ti)-lp
dt, I eiwi 5 *
(f-3)
--co
where the symbol P stands for ‘principal value’. It follows that PR=COS2Ef(f-(YlX-ply)+
sin 2.5 -P rr
4(w) eiw(r--P1y+t)dm.
(9)
The second expression on the right-hand side of (9) can, however be simplified further, to yield PR = cos 2&f(t - (Y~X-P1y)+sin
2~ g(t-alx
--ply),
(10)
where g(u)=iP
Icof(u+4’)d5 5 . -m
(11)
L1.G. Chambers / Total reflection of a sound pulse
249
The transmitted pressure field will be given by m
101
I
pT=2cOs&
iWr+ie-+]til&,y w
--a3
I
do.
(12)
From the convolution theorem for Fourier Integrals, it follows that 00 P==2COSE
I
_m f(7
(13)
- SM (5) dS,
where (14)
=eieI
0
co
--co
=
exp(w(i5 +&y )} do + e-”
e”(-i[
-
pZly
)-’ + e-“(it
- &,y
I
exp{w (il
--oo
)-l
-
PZ,Y )I dw.
(15)
-1 &,y cos E + 5 sin E
(16)
C-n
Thus the transmitted pressure field is given by m
-2lr-i
I
cos & _J(l-W5)
P2,y cos E + 5 sin E dS. I
(P2rY I2 + c2
I
(17)
3. Fluid particle velocity For an acoustic disturbance, the pressure and the fluid particle velocity V, are related through a velocity potential 4, by the relations [5] v=vq5, p = -p
(18)
aq5/dt
(19)
av/at.
(20)
giving the result Vp = -p
This is of course the linearized equation of motion of the fluid. The two components of velocity will obey the relations
ap/ax= -paV,/dt
(204
aplay= -pW,/dt.
Gob)
and
L1.G. Chambers / Total reflection of a sound pulse
250
Consider first the disturbance in the half space y > 0. The total pressure field is given by (21)
~~=f(t-culx+P~y)+cos2ef(t-cw~x-Ply)+sin2~g(t-(~1X-B1y). It follows without any difficulty that vx
=
(~lIPl)Pl
(22)
and that
V,=~P~lp~~[co~~~f~t-(~~~-P~~~+s~~~~g~t-~~~-P~y~-f~~-~~x+P~y~l. In the half space y < 0, the pressure associated with the transmitted PT = -26’
field is given by
f0 -
cos &
(23)
(24)
It follows without any difficulty that vx
=
(25)
bl/P2)PT.
The calculation of V, is somewhat more complicated.
av, 1 -_=--at
p2
We have
aP
ay
=-
(26)
Now
a G [
P2,y cos F + [ sin F cos E k’- @2rY)21-2P2ry sin e I =P2r KP2rY)2+~212 @2rY12+C2 a
p2ry sin
E - 5 cos F I (P2,y)2+52
=%J Thus
?.5 2&!.E np2
co
I
f(t-cr1x-O-
--oD
a
1
&y sin F -C cos 8 dl.
a6 [
032rY)2+L2
Integrate by parts, and it follows that
7$= 2p2,z2 w2
because, provided
m
I
f'(t--1x
--oo
-5)
1
P2,y sin E - 5 cos E d5,
[
032rY)2+52
f is always finite, the integrated part vanishes at the limits. Thus (27)
251
LLG. Chambers / Total reflection of a sound pulse
4. Energy flow
The energy flow E for an acoustic disturbance is given by the equation E =pV.
(28)
The boundary conditions at an interface, which were in fact used in the derivation of the expressions (2) and (3), are that both the pressure, and the normal component of velocity, are continuous across the interface. It follows therefore that the normal component of energy flux across the interface will be continuous. This quantity is given by pV,. As the expressions for p and V, in y > 0 are slightly simpler, it will be easier to calculate the energy flux across the interface from these. For y>O, E,=[f(t-cvlx+~ly)+cos2~f(t-~1~-~ly)+sin2~g(t-culx-~ly)] (2%
x(Pl/pl)[Cos2Ef(f-(Y1X-PlY)+Sin2Eg(f-CYlX-P1Y)-f(t-(Y1X+PlY)l.
The flux across the interface is, on putting y = 0, therefore given by (&/pi)[f{l
+cos 2E}+ g sin 2e][f(cos 2~ - 1) + g sin 283 =
= (P1lP&?
-f’) sin22e +fg sin 4~).
(30)
The net energy flux over all time across the interface at a particular point is therefore {[go - a1x)]2-[f(t-~1~)]2}dt+sin4~
Now it is a well-known result that [2] 03 _m {f(t) +ig(t)}’ dt = 0. I
dt
. I
(31)
(32)
Thus it follows that the net energy flux over all time at a particular point is zero. In exactly the same way, it follows by integrating with respect to x that the net energy flux across the surface at a particular time is zero.
5. Some examples
(a) If f(t) = cos wt, we have (33) It follows from this that an incident pressure wave given by cos w(t--six
+&y)
(34)
will give rise to a reflected pressure wave of the form cos[w(r-aix-ply)+2&].
(35)
Expressions (34) and (35) are in fact the real parts of the expressions (1) and (2) respectively. Similarly, it may be shown that the correct expression is obtained for the pressure of the transmitted disturbance. For the
L1.G. Chambers / Total reflection of a sound pulse
252
harmonic field, the energy flux across the interface is given by (p1/p>[{cos2 w7 -sir? wr} sin2 2s - cos 07 sin w7 sin 4~1.
(36)
It can easily be verified that the mean value of this over a time (2n/w) or over a strip xo
(37)
and formally (38) This step can in fact be made legitimate by taking s(t) as the limit of some sequence. The reflected pressure field will be pn=S(t-(~i~-~iy)cos2~-sin2~/{t-~~~-/3~y}.
(39)
It follows from (17), that the associated transmitted pn= -26’
cos & (&ry cos &+ (t -six)
pressure field is given by
sin .5)/{p2,y)2 + (t-(Y1X)2}.
(40)
Inspection of (39) and (40) reveals that the reflected and transmitted fluid are ahead of the incident pulse on the interface, and the transmitted and reflected field exist for all time, even when the incident pulse has a well defined front. Although this phenomenon is apparently rather suprising, the reason for it is quite straightforward. The velocity of propagation of the incident pulse, parallel to the interface is C1 sin &, whereas the velocity of propagation in the reflecting medium is C2. Thus the transmitted disturbance in the second medium will run along the interface ahead of the incident disturbance and will therefore cause a reflected disturbance which will also run ahead of the incident pulse. This is because of the condition for total internal reflection C2 > Ci cosec 13~.
(41)
f(t) = H(t) cos(wt + y),
(42)
(c) If
where H(t) is the Heaviside step function, we have g(u)=$P
O”H(u +~)cos(ou +w<+y) I -cc
= 1 cos(wu + y) P 7r
r
dl Z-P1 m cos(wu + or + y) dl IT I --oo s (wu +y) P
(44)
Now
P
(43)
ifu>O_
L1.G. Chambers / Total rejection of a sound puke
Alternatively P
253
if u < 0, the integrand is non singular over the range of integration and so it is possible to write *
J--u
6,ydl=
ydl=
-Cilouj.
Also P
msin w 5 dl = -si(-wu). I-I4 5
Thus g(u) -- f sin(wu + v)si(-wu)-i
(45)
cos(wu + y)Cilwu).
Similar comments to those given after (40) apply here. It will be noted that there will be a singularity in g(u) at u =O. The expression (17) for the transmitted pressure field can also be evaluated but the process is rather long and complicated and will be omitted here. (d) If f(t) = H(t)
-H(t - r*), r* > 0,
we have
(46) In exactly the same way, the transmitted pressure field is given by -26’
=
cos E
7*-1p2,y cos t + 5 sin I --T &rY12+t2
-27r-’ cos 8
Cf
t dy =
‘*PT~os tan-l (&)
= TT-’ cos2 E[tai’fg)
-tan-l(&)]
+ d sin logk2 + @2ry)zl]
+r-lCOS
E
sin 8 log[ (7 -‘::,!~$~,,2].
(47)
References [1] A.B. Arons and D.R. Yennie, “Phase distortion of acoustic pulses obliquely reflected from a medium of higher sound velocity”, _T.Acoust. Sot. Amer. 22, 231-237 (1950). [2] P.L. Butzer and R.J. Messel, Fourier Analysis and Approximation, Vol. 1: One Dimensional Theory, Birkhauser, Verlag Base1 (1971) 316. [3] W.M. Ewing, W.S. Jardetsky and F. Press, Elastic Waves in Layered Media, McGraw-Hill, New York (1957) 90. (1945). [4] F.G. Friedlander, “On the total reflection of plane waves”, Quart. .I. Mech. Appl. Math. 1,376-383 [SJ L.D. Landau and E.M. Lifschitz, Fluid Mechanics, Pergamon, London (1959) 246. [6] J.W.S. Rayleigh, The Theory of Sound Vol. 2, Dover, New York (1945) 83.