Wave MotionWave Motion"And your head is shaking
and your arms are shaking
and your feet are shaking,
cause the Earth is shaking"
- REM
Lecture Outline:
There are many wave type motions. Ocean waves, sound waves, light waves, radio waves, waves on strings. The types of waves we will talk about today are mechanical waves. These are waves that move through some physical medium like air, water, strings, earth, etc. Light and radio waves are not mechanical waves; they are waves in the electromagnetic field. While electromagnetic waves obey many of the same principles as mechanical waves they do have some important differences so we will leave the discussion of electromagnetic waves until next semester.
But what exactly is a wave? Einstein and Infield, in their book The Evolution of Physics, describe a wave in this way: "The wind passing over a field of grain, sets up a wave which spreads out across the whole field. Here again we must distinguish between the motion of the wave and the motion of separate plants, which undergo only small oscillations....The essentially new thing here is that for the first time we consider the motion of something which is not matter, but energy propagating through matter."
A mechanical wave is a disturbance moving through a medium.
A. A wave pulse is generated by shaking one end of a stretched string. The pulse travels to the other end of the string. It carries with it
Waves are identified by the relationship between the motion of the disturbance and the motion of the medium. Transverse waves are waves in which the motion of the particles in the medium is perpendicular to the direction of motion of the disturbance. Waves on a string are transverse waves.
Longitudinal waves are waves in which the motion of the particles in the medium is in the same (or opposite) direction as the motion of the disturbance. Sound waves are longitudinal waves. In sound the particles of the medium move back and forth creating regions of high and low density (or high or low pressure).
Linear mechanical waves have obey the superposition principle:
If two or more traveling waves are moving through a medium, the resultant wave function at any point is the algebraic sum of the wave functions of individual waves.
There are two important consequences of the superposition principle. The first is that two waves can pass through each other with being altered. (Notice how this is different from the motion of particles.) The second is that when two waves are in the same region they interfere with each other and produce a pattern different from either wave alone.
The two most important situations in wave interference are called constructive and destructive interference. When two identical waves moving toward each other meet they add together to produce a wave double the amplitude of either - this is called constructive interference.
and 
If two waves that are identical except that one is inverted with respect to the other meet they add together and cancel each other out. (Once they have passed each other they reappear unaltered.) This is destuctive interference
In general waves combine according to the superposition principle:
B. Two identical wave pulses with opposite amplitude travel toward each other on a string. When they interfere destructively
When a wave reaches a boundary it may be reflected. There are two important cases: the fixed boundary and the moveable boundary. If the end of a string is fixed a pulse will be reflected back the way it came but it is inverted. If the end of the string is free to move the pulse will be reflected unchanged.
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A wave can also be reflected when it moves from one medium to another. When a wave pulse strikes a boundary between mediums with different wave velocities some of the wave is reflected and some is transmitted. When a wave pulse travels from medium A to medium B and the wave speed in A is greater than the wave speed in B, the pulse is inverted upon reflection. If the wave speed in A is less than the wave speed in B the pulse is not inverted upon reflection.
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The speed of a wave is determined by the medium. In general the speed of the wave can be thought of as being dependent on two properties of the medium
For a string the speed is given by
where T is the tension in the string and m is the mass/ lengyh of the string.
The speed of sound waves in a medium depends on the compressibility and mass density of the medium. The speed of sound can be calculated from the bulk modulus, B, and the density r. (The bulk modulus is a measure of resistance to change in volume of a solid or liquid.
The speed of sound in air is about 343 at 20 degrees C and 1 atm pressure. As the temperature increases the speed of sound in air tends to increase according to
C. It the Earth's gravity was somehow increased, ocean waves would travel
D. While watching a thunderstorm you see a flash of lightening and then, 4 seconds later, you hear the crack of thunder. Roughly how far is the thunderstorm from you?
E. Which of the following substances would you expect to have the largest speed of sound?
An important type of wave is the sinusoidal wave. Sinusoidal waves are contious waves described by sine functions.
For a sinusoidal wave moving to the right the equation is
For a wave moving to the left the equation is
where the wave number, k, is defined as
The wavelength, l, is the distance between succesive crests. Below is a"snapshot" graph showing the position of the wave at a given instant in time with the wavelength indicated.
In addition to the "snapshot" graph we can draw a "history" graph that shows the movement of a given particle in the medium over time.
The period T is indicated on the graph. Remember the peiod is related to the frequency and angular frequency.
While the examples above have used transverse waves you can use sinusoidal wave equations to describe longitudinal waves as well. The pressure change in a sound wave might be graphed as
For a sinusoidal wave the wavelength and frequency are related to the speed of the wave in the medium.
The speed of the wave is determined by the medium. If the medium through which the wave travels changes there must be a change in the wavelength or frequency.
F. Two strings, one thick and one thin, are connected together to form a single string. A wave travels from the thick string to the thin string. As it passes from one to the other there is a change in the wave's
G. A sound wave in air has a wavelength of 50 cm. What is its frequency?
A. The answer is 3. Both energy and momentum are carried by the wave. The particles of the medium have a velocity as the disturbance passes by. As the particles have mass and velocity they have both momentum and energy.
B. The answer is 2. The string is straight at one point in time but the string is still in motion at that moment.
C. The answer is 1. Consider a small piece of water in the wave. The restoring forces (weight and buoyant forces) increase when gravity increases. The mass of the water remains unchanged. Compare this to the speed of a wave on a string. If the the tension of the string (restoring force) increases and the mass per unit length remains unchanged the speed of the string's waves increases. Hence we would expect the ocean waves to increase in speed also.
D. The answer is 4. Four seconds times 343 m/s gives about 1.3 km.
E. The answer is 5. Steel has a very high bulk modulus compared to its density. This is why it is used for construction of tall buildings.
F. The answer is 4. The frequency is determined by the time of the shaking of the string so it doesn't depend on the thickness. (It depends on whatever is shaking it.) The velocity does depend on the medium so it will change. In order to maintain the relationship between the speed, wavelength and frequency the wavelength must change as well.
G. The answer is 4. Using v = fl and the 343 m/s for the speed of sound we find 686 Hz for the frequency.
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