2. • When sound passes through a medium, there are alternating areas of low
pressure and high pressure due to the oscillations of molecules
• High pressure (Compression) : medium is compressed
• Low Pressure (Rarefraction) : medium is stretched
• The molecules are displaced parallel/antiparallel to the direction of
propagation
SOUND WAVES ARE LONGITUDAL
WAVES
3. • The speed depends on the characteristics of the medium
As you recall from Chapter 14, the speed of waves on springs is determined by the
mass density (μ) and tension T
• v =√T/μ
For air, the mass density is simply the density of the medium ρ
Another flashback to Chapter 14, the spring constant k defines how much the length
changes with force. Bulk modulus B is similar to k as it is defined by how much the
volume changes when the pressure exerted is changed.
B = -(Δp/ΔV)*V
There is a negative sign in this formula because the change in volume is opposite in
sign to the change in pressure
We can take this information and plug it into the equation for the velocity of a wave
on a string and we get the speed of sound as v=√B/ρ
SPEED OF SOUND
4. If the wave is travelling in the positive x direction then
s (x,t)=smcos(kx-ωt+φ)
with sm = maximum displacement from equilibrium,
wave number k= 2π/λ,
Angular frequency ω= 2πf
And the phase constant φ
And
• The relationship between velocity, frequency and wavelength
• v=fλ=ω/k
DISPLACEMENT AMPLITUDE
5. • Each medium has an ambient pressure, and each increase or decrease
relates to a pressure variation
• Displacement variations and pressure variations have the same periodicity
(both in time and space)
• A single frequency sound wave varies sinusoidally
In regions of high pressure, maximum pressure is when the displacement is
equal to zero
Similarly, in regions of low pressure, minimum pressure is when the
displacement is equal to zero
PRESSURE VARIATIONS
6. Change in displacement: Δs = (ϑs/ϑx)*Δx
This is a partial derivative because s depends on both x and t
Change in volume: ΔV=AΔs
Which, from the previous equation, means ΔV=A(ϑs/ϑx)Δx
In the equation we multiply the change in displacement by the cross-sectional
area to find the change in volume and rearrange it to get
(ΔV/AΔx)= ΔV/V = (ϑs/ϑx)
In order to relate the volume variations to pressure variations, we substitute
volume into the bulk modulus formula to achieve
(ΔV/V) = -Δp/B = (ϑs/ϑx) which is algebraically rearranged to
Δp=-B(ϑs/ϑx)
FORMULA DERIVATION
7. By taking the partial derivative of the displacement formula we can achieve
(ϑs/ϑx)= -ksmsin(kx-ωt+ϕ)
And since -B(ϑs/ϑx)=Δp,
Δp=Bksmsin(kx-ωt+ϕ)
CONTINUED
9. The difference between the wave and pressure formulas is that the wave uses a
cosine function and the pressure formula uses a sinusoidal function.
Each function has the same wavelength, period and wave speed except they
are π/2 out of phase from each other
WAVE AND PRESSURE FORMULAS
10. Intensity is the power delivered per unit area
I = P/A
P is the rate at which the wave delivers energy and A is the area the wave is
influencing.
The power of a mechanical wave is described as
Pavg=(1/2)μvω2A2
Sound wave density is represented by ρ and the amplitude is sm, giving us the
equation for the intensity of a sound wave as
I=(1/2)ρvω2sm
2
INTENSITY
11. 1.) Compared to sound of a 44 decibels, a sound of 34 decibels is
a) 5 times less intense
b) 10 times less intense
c) 10 times more intense
d) 100 times less intense
2.) True or False: The bulk modulus is related to the change in pressure but
not the change in volume.
PRACTICE
12. 3.) If this is the displacement graph of a sound wave,
then what is the graph of the pressure change for this wave
PRACTICE
13. 4.) True or False: A in the formula for intensity represents the cross-sectional
area of the wave
5.) Show how the velocity = fλand ω/k
6.) What is the velocity of a wave with a 5 mm wavelength that oscillates three
times completely in one second?
7.) If the change in a volume of 20 cm2 is 12 cm2 and the change in x is 5
cm, what is the change in displacement?
PRACTICE
14. 1.) b: 10 times less intense
2.) False: Δp=-B(ϑs/ϑx) and ΔV/V = (ϑs/ϑx) relating bulk modulus to both
the change in pressure and change in volume
3.) The funcion of the pressure is
π/2 radians out of phase with
the displacement function but
have the same wavelength and
period
ANSWERS
15. 4.) False: A is not the cross-sectional area of the wave itself, but the area that
the wave is affecting.
5.) ω= 2πf and k=2π/λ
If v= ω/k then v = (2πf)/(2π/λ)
v = (2πf)*(λ/2π)
The 2π cancel out and we are left with
v = fλ
6.) If the wave oscillates 3 times in a second, the frequency is 3 Hz.
v= fλ and the wavelength was stated to be 5 mm so the velocity is
v= 3Hz * (5mm/1000mm)= 3Hz*(0.005m) = 1.5e-2 m/s
ANSWERS
16. 7.) ΔV/V= (ϑs/ϑx) and
(ϑs/ϑx)*Δx = Δs
So ΔV/V= 12 cm2/20cm2 =(ϑs/ϑx)
So (ϑs/ϑx) = 0.6
And with the change in x being 5 cm,
0.6*5cm= 3 so Δs=3
ANSWERS
17. Physics for Scientists and Engineers “An Interactive Approach”. J. Iqbal, F.
Mansour, M. Milner-Bolotin, P. Williams and R. Hawkes. Nelson Education Ltd.
2014
SOURCES
