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

Wave - IIWave - 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 show

B p

and thus RT p

vM

using constantpV

3. Intensity

Transverse Waves (String):

P 1

2v 2ym

2

Sound Waves (Longitudinal Waves):

I P

A

1

2v 2sm

2A: area intercepting the sound

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

Sound Intensity for a Point Source

Wavefront area at distance r from the source:

A = 4r2

I Ps

A

Ps

4r2

The Decibel Scale

The sound level is defined as:

10dB logI

I0

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 1

2

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 l

v1

l

v2

Solve for l:

l t

1/ v1 1/ v2

0.120 s

1/(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.