Chapter 17

Waves (II)

Sound Waves

Supersonic Speeds, Shock Waves

Sound Waves

Speed of Sound

Pressure Fluctuation in Sound Waves

Interference

Intensity and Sound Level

Sources of Musical Sound

Beats

Doppler Effect

Key contents:

17.1 Sound Waves

Sound waves are longitudinal waves of density and pressure fluctuations.

# For human ears, audible in 20 Hz ~ 20000 Hz

# In this chapter, we will focus on sound waves in the air.

17.2 Speed of Sound

The wave speed on a string is

The bulk modulus B is defined to be the elastic factor for a gas:

With the density being the inertia factor, the sound speed can be most likely expressed as

17.2 Speed of Sound

We have:

Also,

And

Therefore,

But

Finally,

A derivation:

17.3 Pressure Fluctuation in Sound Waves

}

17.3 Pressure Fluctuation in Sound Waves

# Δs is the displacement change at the two sides of the element Δx, while Δp is the pressure excess in Δx.

# Δp and s are always with a 90o phase difference.

# From , one may also see that

Example, Pressure and Displacement Amplitudes

17.4 Interference

17.4 Interference

Phase difference can be related to path length difference L, by noting that a phase difference of 2 rad corresponds to one wavelength.

Therefore,

Fully constructive interference occurs when is zero, 2, or any integer multipleof 2.

Fully destructive interference occurs when is an odd multiple of :

Example, Interference:

Example, Interference:

17.5 Intensity and Sound Level

The intensity I of a sound wave at a surface is the average rate per unit area at which energy is transferred by the wave through or onto the surface.

Therefore, I =P/A where P is the time rate of energy transfer (the power) of the sound wave and A is the area of the surface intercepting the sound.

The intensity I is related to the displacement amplitude sm of the sound wave by

17.5 Intensity and Sound Level

Consider a thin slice of air of thickness dx, area A, and mass dm,oscillating back and forth as the sound wave passes through it.The kinetic energy dK of the slice of air is

But,

Therefore,

And,

Then the average rate at which kinetic energy is transported is

If the potential energy is carried along with the wave at this same average rate, then the wave intensity I, the average rate per unit area at which energy of both kinds is transmitted by the wave, is

Another derivation: the instantaneous power supplied to the wave element in the following figure is(p is the pressure excess)

17.5 Intensity and Sound Level

This should be compared with that on a string:

s

(recall )

17.5 Intensity and Sound Level: Variation with Distance

17.5 Intensity and Sound Level: The Decibel Scale

Here dB is the abbreviation for decibel, the unit of sound level.

I0 is a standard reference intensity ( 10-12 W/m2), chosen near the lower limit of the human range of hearing.

For I =I0 , =10 log 1 = 0,(our standard reference level corresponds to zero decibels).

# office conversation, 60 dB heavy traffic (3m), 80 dB loud rock music, 120 dB Jet engine (20m), 130 dB

Ex. For sound waves at 1 kHz in the air(density is 1.29 kg/m3, sound speed 340 m/s)

@ 0 dB, s0 = 1.07 x 10-11 m p0 = 2.96 x 10-5 Pa (1 atm = 1.013 x 105 Pa)

@ 120 dB, s0 = 1.07 x 10-5 m p0 = 29.6 Pa

Example, Cylindrical Sound Wave:

Example, Decibel, Sound Level, Change in Intensity:Many veteran rockers suffer from acute hearing damage because of the high sound levels they endured for years while playing music near loudspeakers or listening to music on headphones. Recently, many rockers, began wearing special earplugs to protect their hearing during performances. If an earplug decreases the sound level of the sound waves by 20 dB, what is the ratioof the final intensity If of the waves to their initial intensity Ii?

17.6 Sources of Musical Sound

Musical sounds can be set up by oscillating strings (guitar, piano, violin), membranes (kettledrum, snare drum), air columns (flute, oboe, pipe organ, and the digeridoo of Fig.17-12), wooden blocks or steel bars (marimba, xylophone), and many other oscillating bodies. Most common instruments involve more than a single oscillating part.

17.6 Sources of Musical Sound

A. Pipe open at both endsB. Pipe open at one end only

Example, Double Open and Single Open Pipes:

17.7 Beats

When two sound waves whose frequencies are close, but not the same, are superimposed, a striking variation in the intensity of the resultant sound wave is heard. This is the beat phenomenon. The wavering of intensity occurs at a frequency which is the difference between the two combining frequencies.

Example, Beat Frequencies:

17.8 Doppler Effect

When the motion of detector or source is toward the other, the sign on its speed must give an upward shift in frequency. When the motion of detector or source is away from the other, the sign on its speed must give a downward shift in frequency.

Here the emitted frequency is f, the detected frequency is f’, v is the speed of sound through the air, vD is the detector’s speed relative to the air, and vS is the source’s speed relative to the air.

Example, Doppler Shift:

17.9 Supersonic Speeds, Shock Waves

Homework:

Problems 14, 22, 36, 49, 62