1. Wave - II

2. These are material waves.
1. Sound Waves
Waves on Strings, etc.: Transverse Waves
Sound Waves: ANY Longitudinal Waves
These are material waves.

3. Wave Function s(x,t) = smcos(kx-wt)
y(x,t) = ymsin(kx-wt)
Wave Function
Transverse wave
s(x,t) = smcos(kx-wt)
s: The displacement from the equilibrium position
The sin and cos functions are identical for the wave function, differing only in a phase constant. We use cos in this chapter.
sin(q+90˚)=cosq

4. Pressure Amplitude ∆p(x,t) = ∆pmsin(kx-wt)
∆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

5. 2. Wave Speed Transverse Waves (String):
Bulk modulus
2. Wave Speed
Transverse Waves (String):
Tension
Linear density
elastic
inertial
Sound Waves (Longitudinal Waves):
elastic
inertial
Bulk modulus
Volume density

6. Bulk Modulus

7. 3. Intensity Transverse Waves (String):
Sound Waves (Longitudinal Waves):
A: area intercepting the sound

8. Wavefront, Ray, and Spherical Waves
Wavefront: Equal phase surfaces
Spherical: spherical waves
Planar: 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

9. Sound Intensity for a Point Source
Wavefront area at distance r from the source:
A = 4pr2

10. The Decibel Scale The sound level b is defined as: decibel
10-12 W/m2, human
hearing threshold

11. 4. Interference
For two waves from two different point sources, their phase difference at any given point depends of their PATH LENGTH DIFFERENCE ∆L
x  x+l
kx  kx+2p
f = 0: constructive
f = p: destructive
other: intermediate

12. Constructive:
m=0,1,2, ...
Destructive:
f = m(2p), m=0,1,2, ...
f = 0: constructive
f = p: destructive
other: intermediate
f = (m+1/2)(2p), m=0,1,2, ...

13. 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

14. The time interval between the two sounds:
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 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:
Solve for l:

15. ∆p(x,t) = ∆p msin(kx-wt)
HRW 18P (5th ed.). The pressure in a traveling sound wave is given by the equation
∆p = (1.5 Pa) sin p[(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-wt)
∆p(x,t) = ∆p msin(kx-wt)
(a) ∆pm = 1.5 Pa
(b) f = w/2p =(330 s-1)/2 =165 Hz
(c) l=2p/k = 2p /(1.00 m-1) p=2 m
(d) v = lf =330 m/s

16. HRW 23P (5th ed.). Two point sources of sound waves of identical wavelength l and amplitude are separated by distance D = 2.0l. 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:
(a) Maximum: ∆f=2mp
sinq = m/2
(m=0, ±1, ±2, …)
Eight: 0˚, 30˚, 90˚, 150˚, 180˚,
210˚, 270˚, 330˚
(b) Eight, in between the maximums.