Wave - II
1. Sound Waves
Sound Waves: ANY Longitudinal Waves
Waves on Strings, etc.: Transverse Waves
These are material waves.
Wave Function
s(x,t) = smcos(kx-t)s: The displacement from the equilibrium
position
y(x,t) = ymsin(kx-t)Transverse wave
The sin and cos functions are identical for the wave function, differing only in a phase constant. We use cos in this chapter.
sin(+90˚)=cos
Pressure Amplitude
∆p(x,t) = ∆pmsin(kx-t)
∆p: the pressure change in the medium due to compression (∆p >0) or expansion (∆p <0)
∆p(x,t) and s(x,t) are 90˚ out of phase
2. Wave Speed
Sound Waves (Longitudinal Waves):
v B
Tension
Linear density
elastic
inertial
Bulk modulus
Volume density
elastic
inertial
/pB
V V
Bulk modulus
v
Transverse Waves (String):
Bulk Modulus
/p dpB V
V V dV
one can showB p
and thus RT pvM
using constantpV
3. Intensity
Transverse Waves (String):
P 12v 2ym
2
Sound Waves (Longitudinal Waves):
I PA
12v 2sm
2 A: area intercepting the sound
Wavefront, Ray, and Spherical Waves
Wavefront: Equal phase surfacesSpherical: spherical wavesPlanar: planar waves
Ray: The line wavefront, that indicates the direction of travel of the wavefront
At large radius (far from a point source):spherical wavefront planar wavefront
Sound Intensity for a Point Source
Wavefront area at distance r from the source:
A = 4r2
I PsA
Ps
4r2
The Decibel Scale
The sound level is defined as:
10dB logII0
decibel
10-12 W/m2, human hearing threshold
4. Interference
= 0: constructive = : destructiveother: intermediate
For two waves from two different point sources, their phase difference at any given point depends of their PATH LENGTH DIFFERENCE ∆L
k 2
x x+ kx kx+2
2
L
L
2
1 sinmy t y kx t
2 sinmy t y kx t
L
2
= 0: constructive = : destructiveother: intermediate
= m(2), m=0,1,2, ...
= (m+1/2)(2), m=0,1,2, ...
Destructive: L m 12
Constructive: L mm=0,1,2, ...
Standing Waves in a Tube
BOUNDARY CONDITIONS:
Closed End: s = 0, a node for s
∆p = ∆pm, an antinode for ∆p
Open End: s = sm, an antinode for s
∆p = 0, a node for ∆p
HRW 9P (5th ed.). A man strikes a long aluminum rod at one end. A woman at the other end with her ear close to to the rod, hears the sound of the blow twice (once through air and once through the rod), with a 0.120 s interval between. How long is the rod?
Let the length of the rod be l, the speed of sound in air be v1, and the speed of sound in the rod be v2.
The time interval between the two sounds:
t lv1
l
v2Solve for l:
l t
1/ v1 1/ v2
0.120 s1/(343 m/s) -1/(6420 m/s)
43.5 m
HRW 18P (5th ed.). The pressure in a traveling sound wave is given by the equation
∆p = (1.5 Pa) sin [(1.00 m-1)x - (330 s-1)t].Find (a) the pressure amplitude, (b) the frequency, (c) the wavelength, and (d) the speed of the wave.
s(x,t) = smcos(kx-t)
∆p(x,t) = ∆p msin(kx-t)
(a) ∆pm = 1.5 Pa
(b) f = /2=(330 s-1)/2 =165 Hz
(c) =2/k = 2 /(1.00 m-1) =2 m
(d) v = f =330 m/s
HRW 23P (5th ed.). Two point sources of sound waves of identical wavelength and amplitude are separated by distance D = 2.0. The sources are in phase. (a) How many points of maximum signal lie along a large circle around the sources? (b) How many points of minimum signal?
The phase difference at point P:
2x
2
Dsin 4 sin
(a) Maximum: ∆=2msin= m/2
(m=0, ±1, ±2, …)Eight: 0˚, 30˚, 90˚, 150˚, 180˚,
210˚, 270˚, 330˚
(b) Eight, in between the maximums.