A Review of some properties of Sound

Speed of sound in different materials…

1. V = (B/r)^(1/2)

Where B = bulk modulus and r = density (mass/volume)

Material	Bulk Modulus	Density
Iron, cast	90 x 10^9 N/m^2	7.8 x 10^3 kg/m^3
Steel	140 x 10^9 N/m^2	7.8 x 10^3 kg/m^3
Granite	45 x 10^9 N/m^2	2.7 x 10^3 kg/m^3
Water	2.0 x 10^9 N/m^2	1.0 x 10^3 kg/m^3
Alcohol	1.0 x 10^9 N/m^2	0.79 x 10^3 kg/m^3

Ques: Calculate the speed of sound waves (these are longitudinal waves) in Granite and in Alcohol…

Ans: look up the bulk modulus and density for each of these substances…

V_granite = (45 x 10⁹ N/m²/2.7 x 10³ kg/m³)^(1/2)

V_granite = 4082 m/s

You can try alcohol using the table info above….

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2. Intensity (decibels - dB)

a)Find the decibel level for an intensity of 4.5 x 10^-6 watts/m²?

Use b = 10 log(I/[1 x 10^-12])

b = 10 log([4.5 x 10^-6]/[1 x 10^-12])

b = 10 log(4.5 x 10⁶)

b = 10 (6.65)

b = 66.5 dB

b)A decibel level of 75 dB is measured. What intensity is this?

75 dB = 10 log(I/[1 x 10^-12])

7.5 = log(I/[1 x 10^-12])

10^7.5 = (I/[1 x 10^-12])

(1 x 10^-12) 10^7.5 = I

I = 3.16 x 10^-5 W/m²

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3. Doppler Effect…(or, I was going so fast, Officer, I thought the traffic light was still yellow)

Well, there's just a ton of variations for this. Let's use the general equation and take the example of a train approaching the station at 25 m/s while a passenger is running towards the station (and the train) at 5 m/s. The train whistle is at an annoying 750 Hz.

f' = f[ (1 +- v_o/v) / (1 -+ v_s/v) ]

here v_o = observer velocity, v_s = source velocity, v = speed of sound (which we'll assume is 340 m/s). The first sign is for motion towards the object, the second sign is for motion away.

So, armed with that knowledge both are towards each other)

f' = (750)[ (1 + 5/340) / (1 - 25/340) ]

f' = (750)[ 1.147 / 0.927 ]

f' = 928 Hz

Of course, you can also have mixed cases, such as the observer moving away from the source while the source approaches him. In that case you would use the appropriate sign (upper for towards, lower for away) corresponding to V_o (observer) or V_s (source).

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4. Standing waves.....

A standing wave can be described as a fixed (resonant) vibration pattern on a string or in a tube.

For now we'll talk about tubes: at the ends of a tube the molecules are free to vibrate at maximum amplitude. So imagine a tube: at the center, there is no molecule motion, but at the ends, the molecules oscillate back and forth the maximum amplitude.

So a tube open at both ends has possible amplitude vibrations that looks like this:

The dark regions are hi pressure regions (antinodes)

(amplitude means the air molecules are oscillating sideways: a node is where the air molecule doesn't move at all)…

Now a tube with one end closed, has vibrations that look like:

Notice the trend there: l = 4L/n where n = only odd numbers!!

Okay, now for some examples…

Let's do one tough one:

Suppose you know that two frequencies 1760 Hz and 1320 Hz are present on a 70 cm tube open at both ends (with no frequencies in between). A)what's the fundamental? B) what's the speed of the wave?

a) OK, with a tube open at both ends, all integer harmonics are present so 1320 represents "n" and 1760 represents "n+1". The difference is just the fundamental!

Ans: 440 Hz

b) the speed of the wave is given by,

f_n = nV/2L

440 Hz = (1)V/2(0.70 m)

and solve for V….

Good Luck!