Summary of Chapter 18

 

1.     Sound waves are longitudinal; that is the particles of the medium move parallel to the direction of motion of the wave.

 

The velocity of sound in different media is given by;

 where Y is the Young modulus

 where B is the Bulk modulus

v_air = 343 m/s at a temperature of about 20 ^oC and v_vacuum = 0

 

2.     A harmonic sound wave can be described by a displacement wave or a pressure wave.

 

The displacement wave for a harmonic sound wave is given by;

where s(x,t) is the diplacement of the particles in the medium, k is the wave

number, w is the angular frequency, and S_m is the displacement amplitude.

 

The pressure wave is given by;

Where

DP_m is the pressure amplitude and r is the density of the medium and v is te speed of sound in the medium.

 

3.     Interference of sound waves  

 

The relationship between the difference in path and the phase difference between the two sound waves at the location of a listener is

 

 

DL is the path length difference between the two sound waves.

 

2 Cases:

 

a)                Constructive interference (maximum sound) DL = 0, l, 2l, 3l

 

Þ                for  n = 0, 1, 2, 3, 4,

 

 

b)                Destructive interference (minimum sound) Dr = 0, l/2, l, 3l/2 .

 

Þ              for  n = 1, 3, 5, 7,

 

 

4.     The power transmitted in a harmonic sound wave is given by;

The intensity of a sound wave I is defined as          (W/m²)

Þ    

Since the intensity of sound varies between 10^-12 W/m² to 1 W/m² we define a new quantity called sound intensity level b as

 

where I_o = 10^-12 W.m² is the reference intensity.

The units for b is dB (Decibel). Now b varies between 0 and 120 dB.

 

For spherical sound  waves, the intensity is given by;

(r: distance between the source and the point where we want to measure the intensity).

 

  and             Þ      

 

5. Standing Waves in air columns (pipes)

 

Sound sources can be used to produce longitudinal standing waves in air columns.

 

2 Cases:

 

a)                 pipe open at both ends: The resonances occur for

 

L = n l/2               for n = 1, 2, 3, 4, ..

 

since v = lf                    Þ     f = (n v)/2L    for n = 1, 2, 3, 4, ..

 

where v is the speed of sound waves

 

b)                pipe closed at one end: the resonances occur when

 

L = n l/4               for  n = 1, 3, 5, 7,

 

Þ     f = n v/4L                  for n = 1, 3, 5, 7,

 

 

6.     The Doppler Effect

The Doppler effect is the change in frequency f heard by a detector whenever there is  relative motion between a source and a detector. There are 8 cases described as follows:

 

Detector      Source                  Equation

D                     S             Detector moving toward stationary source

D                 S                 Detector moving away from stationary source

D                 S                 Source moving away from a stationary  detector

D                 S                 Source moving toward a stationary detector

 

D                 S                 Detector approaching and source is moving away

 

D                 S                 Source approaching and detector moving away

 

 

D                 S                 Source and detector are both approaching

 

D                 S                 Source and detector are both moving away

InGeneral