Dr. Johnny B. Holmes

Vibrations and Waves
Sound
Return to PHYS 201 Outline

Outline
Supplementary Homework Problems
Answers to Problems
Return to top

Outline

1. Hooke's Law: F_S = kx where k is the spring constant
1. Newton's Second Law: S F = ma leads to -kx = ma
2. Conservation of Energy: PE = 1/2kx² leads to 1/2mv² + 1/2kx² = E
3. oscillations (solution of differential equation: -kx = m d²x/dt² gives)
1. x = A sin(w t+q )
2. frequency of oscillations: f = w /2p ; w = Ö(k/m)
3. period of oscillations: T = 1/f
4. amplitude: A (related to energy)
2. The pendulum YY
3. Damped oscillations
4. Waves on a string ZZ
1. Newton's Second Law: leads to y(t) = A sin(kx ± wt)
2. velocity of a pulse: v = Ö(T_ension/m ) where m = m/L
3. sine and cosine waves
1. frequency: f = w / 2p ; T = 1 / f
2. wavelength: k = 2p / l
3. relations: v = l / T = l f = w / k
4. reflections of waves
1. standing waves: #(l/2) = L
2. guitar notes: due to standing wave resonances

Return to top of section 1

LETTER PROBLEMS:

YY. Design a pendulum that will give a period of 1.0 seconds.

ZZ. A certain guitar string has a length of 0.9 meters and a mass of 2.1 grams (over the length of 0.9 meters). A tension of 24 Nt. is applied to it. a) What will be the wavelength of the fundamental mode of oscillation when the string is plucked? b) What will be the fundamental frequency when it is plucked? c) What will be the speed of the wave on the string?

Return to top of section 1

YY. any mass, any angle, length = 25 cm.

ZZ. 1.8 meters; b) 56.34 Hz; c) 101.42 m/s.

Return to top of section 1

Outline
Supplementary Homework Problems
Answers to Problems
Return to top

Outline

1. Sound pulses and the speed of sound AAA
   in air: v = Ö(B/r ) = Ö(g RT/M) (where g =C_P/C_V = 1.4 for air)
2. Sound pulses and sine waves: Fourier analysis
   The ear is a good fourier analyzer for sound, but the eye for light is NOT!
3. Intensity of sound
1. I_ntensity = P_ower/A_rea (in Watts/m²)
2. logarithmic scale: decibels: I(db) = 10 log(I/I_o)
   range of human ear: for I [10^-12 W/m² to 1 W/m² ],
   and for f [20 Hz to 20000 Hz]
3. P_{ower µ} A² and P_{ower µ} f², therefore same is true for I
   effect on speakers: higher f requires lower A for same I
4. Plane and spherical waves BBB
1. plane waves: uniform intensity
2. point sources: inverse square law: I = P_avg / 4p r² (no absorption)
5. Doppler effect CCC
1. speed of sound relative to medium
2. frequency of sound relative to source, receiver, and medium:
   f_Receiver = f_Source [v_medium ± v_Receiver] / [v_medium ± v_Source]
6. Resonance

Return to top of section 2

LETTER PROBLEMS:

AAA. a) What is the speed of sound in air in a refrigerated room where the temperature is 0°F ? b) What is the wavelength for a sound of frequency 60 Hz in this air?

BBB. a) In Watts/m², what is the intensity of a sound wave at the 83 dB level? b) In dB, what is the intensity level of a sound wave at 2.7 x 10^-3Watts/m² ? c) If the intensity in part b were doubled to 5.4 x 10^-3W/m², what would the intensity level in dB become? d) If the source of the sound were a point source, and if the point source moved four times further away, what would the intensity level do?

CCC. A train moving East at a speed of 15 m/s approached a person in a car moving West at a speed of 24 m/s. a) If the train emits a sound of frequency 1000 Hz, what will the car observer measure for the frequency? (Assume no wind.) b) After the train and car pass and start heading away from each other, what will the car observer measure for the frequency? (Again assume no wind.) c) If there is a wind of 20 m/s blowing from the East (that is, blowing toward the West which is in the opposite direction the train is heading), would the answer to part b increase, stay the same, or decrease?

Return to top of section 2

ANSWERS TO LETTER PROBLEMS:

AAA. a) 325 m/s; b) 5.42 meters.

BBB. a) 2 x 10^-4W/m²; b) 94.3 dB; c) 97.3 dB; d) decrease by a factor of 16 in W/m², and decrease by 12 dB.

CCC. a) 1118 HZ; b) 891 Hz; c) increase (897 Hz).

Return to top of section 2