VIBRATIONS AND WAVE MOTION

Dr. Johnny B. Holmes

## Vibrations and Waves

1. Hooke's Law: FS = 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(Tension/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

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

ZZ. A certain guitar string has a length of 0.7 meters and a linear mass density of 1.2 grams/meter. 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?

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

ZZ. 1.4 meters; b) 101 Hz; c) 141.4 m/s.

## Sound

1. Sound pulses and the speed of sound    AAA
in air: v = Ö(B/r ) = Ö(g RT/M) (where g =CP/CV = 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. Intensity = Power/Area (in Watts/m²)
2. logarithmic scale: decibels: I(db) = 10 log(I/Io)
range of human ear: for I [10-12 W/m2 to 1 W/m2 ],
and for f [20 Hz to 20000 Hz]
3. Power µ A² and Power 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 = Pavg / 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:
6. Resonance

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?