Elasticity, Oscillations, Waves, and Sound
Mechanical Waves
A progressive or traveling wave is a self-sustaining disturbance of a medium that propagates from one region to another, carrying energy and momentum.
Longitudinal Wave – the sustaining medium is displaced parallel to the direction of propagation.
Transverse Wave – the sustaining medium is displaced perpendicular to the direction of propagation.
Period – the time it takes for a wave to go through one complete oscillation.
Frequency – the number of complete oscillations the wave makes in a second.
Wavelength – the spatial distance over which the wave makes one complete oscillation.
Speed of any progressive periodic wave:
(16) 
Example 16
Sound waves travel in air with a speed of 343 m/s. The lowest frequency sound we can hear is 20.0 Hz; the highest frequency is 20.0 kHz. Find the wavelength of sound for frequencies of 20.0 Hz and 20.0 kHz.
Transverse Waves: Strings
Speed of a transverse wave in a string:
(17) 
where 
is
the tension in the string and m/L is the mass per unit length.
Example 17
A 5.0 m length of rope, with a mass of 0.52 kg, is pulled taut with a tension of 46 N. Find the speed of waves on the rope.
Example 18
A 12 m rope is pulled tight with a tension of 92 N. When one end of the rope is given a "thunk" it takes 0.45 s for the disturbance to propagate to the other end. What is the mass of the rope?
Reflection
The reflection of a pulse on a rope with a fixed end point. As the pulse arrives, it exerts a vertical force on the fixed anchor point, which in turn exerts an equal and opposite force on the string. When the string tugs up, the anchor point tugs down. This downward force on the rope generates an upside-down reflected pulse traveling in the opposite direction.
The reflection of a pulse on a rope with a free end point. That free end rises until all the energy of the end segment is stored elastically. It comes to rest at a maximum vertical displacement of twice the height of the crest. Carried up by its inertia, the end segment pulls upward on the rope, generating a reflected wavepulse that travels back toward the source, right side up and simply reversed.
Transmitted
The gray rope has a greater linear mass-density. Reflects like it is reflected from a solid wall.
The gray rope has a greater linear mass-density. Reflects like it is reflected from a non solid wall.
Compression Waves
Superposition of Waves
Superposition Principle – In the region where two or more waves overlap, the resultant is the algebraic sum of the various contributions at each point.
Example 19
Two speakers separated by a distance of 4.30 m emit sound of frequency 221 Hz. The speakers are in phase with one another. A person listens from a location 2.80 m directly in front of one of the speakers. Does the person hear constructive or destructive interference?
Example 20
Two speakers are opposite in phase. They are separated by a distance of 5.20 m and emit sound with a frequency of 104 Hz. A person stands 3.00 m in front of the speakers and 1.30 m to one side of the center line between them. What type of interference occurs at the person's location?
Sound Intensity
The Intensity (I) of a wave is the average energy delivered per unit area per unit time. Or the average power divided by the perpendicular area.
(18) 
Units: 
Example 21
A loudspeaker puts our 0.15 W of sound through a square area 2.0 m on each side. What is the intensity of this sound?
Example 22
Two people
relaxing on a deck listen to a warbler sing. One person, only 1.00 m
from the bird, hears the sound with an intensity of 
. (a) What
intensity is heard by the second person, who is 4.25 m from the bird?
Assume that no reflected sound is heard by either person. (b) What
is the power output of the bird's song?
Intensity-Level
(19) 
This equation gives you the decibel level (dB) of a sound.
Note: To double the loudness of a source, its intensity must be increased by a factor of ten. An increase of 10 dB in sound-level corresponds to a sound that's twice as loud.
Example 23
A crying child
emits sound with an intensity of 
.
Find (a) the intensity level in decibels for the child's
sounds, and (b) the intensity level this child and its
twin, both crying with identical intensities.
Beats
Beats occur when waves of slightly different frequencies interfere.
(20) 
Standing Waves
Standing waves are formed when two waves having the same frequency, amplitude, and wavelength travel in opposite directions through a medium.
Standing waves in a string:
(21) 
(22) 
Example 24
One of the harmonics on a string 1.30 m long has a frequency of 15.60 Hz. The next higher harmonic has a frequency of 23.40 Hz. Find (a) the fundamental frequency, and (b) the speed of waves on this string.
Standing Waves in Air Columns
Chamber -
both ends closed (23)
Chamber - one end open
(24) 
(25) 
Where N = 1, 3, 5,...
Chamber - both ends open
(26)
(27) 
Where N = 1, 2, 3, ....
Example 25
An empty pop bottle is to be used as a musical instrument in a band. In order to be tuned properly, the fundamental frequency of the bottle must be 440.0 Hz. If the bottle is 26.0 cm tall, how high should it be filled with water to produce the desired frequency?
Example 26
An experimental way to tune the pop bottle from the above example is to compare its frequency with that of a 440.0 Hz tuning fork. Initially, a beat frequency of 4 Hz is heard. As a small amount of water is added to that already present, the beat frequency increases steadily to 5 Hz. What were the initial and final frequencies of the bottle?
The Doppler Effect
The Doppler Effect is the change in frequency due to relative motion between a source and a receiver.
The source moving:
(28) 
Use the minus sign when the source is moving toward the observer and the positive sign when the source is moving away from the observer.
The observer moving:
(29) 
Use the plus sign when the observer is moving toward the source and the negative sign when the observer is moving away from the source.
When both the source and the observer are moving:
(30) 
In the numerator,
the + sign corresponds to the case in which
the oberver moves in the direction of the source,
whereas the –
sign indicates motion in the opposite direction.
In the denominator, the – sign corresponds
to the case in which the source moves in the direction
of the observer, whereas the + sign indicates
motion in the opposite direction.
Example 27
A street musician sounds the A string of his violin, producing a tone of 440 Hz. What frequency does a bicyclist hear as he (a) approaches and (b) recedes from the musician with a speed of 11.0 m/s?
Example 28
A train sounds its whistle as it approaches a tunnel in a cliff. The whistle produces a tone of 650.0 Hz, and the train travels with a speed of 21.2 m/s. (a) Find the frequency heard by an observer standing near the tunnel entrance. (b) The sound from the whistle reflects from the cliff back to the engineer in the train. What frequency does the engineer hear?
Example 29
A car moving at 18 m/s sounds its 550 Hz horn. A bicyclist on the sidewalk, moving with a speed of 7.2 m/s, approaches the car. What frequency is heard by the bicyclist?
The End
