Physics

Vandebilt Catholic High School

W. Dupre

Waves

Wave

- traveling disturbance that transmits energy without transferring matter

Types of waves:

1. Electromagnetic
• Examples -- light waves, radio waves, microwaves, X-rays, etc.
• Do not require a medium for transfer; can be transferred through a vacuum
2. Mechanical
• Examples -- sound waves, water waves, etc.
• Require medium for transfer; cannot be transferred through a vacuum
• The speed of the wave depends upon the mechanical properties of the medium.
• Some waves are periodic (particles undergo back and forth displacement as in a sound wave.)
• Some waves are sinusoidal (paricles undergo up and down displacement as in a wave on a string.)

Types of mechanical waves:

1. Transverse

As a wave passes through a point, the particles vibrate at right angles to the direction in which the wave is moving

Notice -- as the wave moves through point A, the particle does not return to its original position until point E

Crest

upward displacement of transverse wave

Trough

downward displacement of transverse wave

2. Longitudinal

As wave passes through a point, the particles vibrate parallel to the direction in which the wave is moving

Wave terms:

1. Wavelength

the distance between two successive in-phase points; symbol is l and SI unit is meters

2. In Phase

Points on a wave that are in the same orientation. In the wave below points “A” & “F” or in phase. So are points “C” and “H”

3. Out of Phase

Points in a wave that are not in the same orientation. In the wave above points “A”, “B”, “C”, “D”, & “E” are out of phase with points “A” & “D” are 180^O Out of phase.

4. Amplitude

maximum displacement of wave; measure of wave's energy

5. Pulse

single disturbance of a medium

6. Frequency

the number of waves passing a point per second; symbol is f and SI unit is Hertz (Hz)

7. Period

time for one wave; symbol is T and SI unit is second

T = 1/f

f = 1/T

8. Speed

the speed with which the wave moves through the medium is dependent upon the properties of the medium only. It is determined by the ratio of the elastic properties of the medium to the inertial properties of the medium. The speed can be calculated by finding the product of the wavelength and the frequency; SI unit is m/s

v = l n     or    v = l f

Wave behaviors

1. When a wave approaches a boundary between two different media three things may happen.

a. The wave can completely transmit into the new medium. This will only occur if the two media’s properties are exactly alike.

b. The wave can completely reflect back into the old medium. This will happen if the two media’s properties are exactly opposite of each other.

c. The wave will partially transmit into the new medium and partially reflect back into the old medium. The amount of transmission and reflection will be determined by the degree of difference between the two media.

1) The more alike the two media the more the wave will transmit.

2) The greater the difference the greater the reflection.

2. The transmitted wave will always remain upright (in its same orientation)

3. The reflected will:

a. remain upright (in the same orientation) if the wave is traveling from a more dense medium to a less dense medium.

b.

Invert (flip orientations) if the wave is traveling from a less dense to a more dense medium.

1. Interference    - result of the superposition of two or more waves
• Superposition Principle - when two waves are in the same place at the same time, (only waves exhibit this behavior, particles cannot do this)the displacement caused by the waves is the algebraic sum of the two waves

• Constructive interference

waves interfere, "adding" to produce a larger wave

• Destructive interference

waves interfere, "adding" to produce a smaller wave

• Standing wave

two waves with the same wavelength, the same frequency, and the same amplitude that are traveling through a medium in opposite directions interfere producing a standing wave

Node

point of zero displacement on a standing wave (destructive interference)

Antinode

point of maximum displacement on a standing wave (constructive interference)

Sound

Characteristics of sound:

• Sound is a longitudinal wave—it cannot travel through a vacuum; it consists of compressions and rarefactions; the louder the sound, the greater its amplitude
• Speed of sound is temperature dependent

at 0°C, speed of sound is 331.5 m/s

rate of change of speed is [0.6 m/s]x[temp]

Sound exhibits wave properties—it reflects producing an echo; it interferes constructively and destructively; it refracts or bends; it diffracts, or spreads around barriers

• Since sound is a longitudinal wave, the particles of the medium are displaced parallel to the direction of the wave.
• The speed of the sound wave is dependent upon the ratio of the elastic properties of the medium to the inertial properties of the medium

Terms:

speed

The speed of a wave is given by v = l f

pitch

frequency; the range of human hearing is from 20 Hz to 20 kHz. Frequencies less than 20 Hz are called infrasonic. Frequencies more than 20 kHz are called ultrasonic.

The pitch is determined by the mass of the vibrating source. The greater the mass that is vibrating the lower the pitch.

loudness

amplitude (energy of a sound wave)

determined by three factors:

1) amount of input energy (work done to create the sound wave)

2) the distance from the source (there is an inverse quadratic relationship between distance from the source and loudness)

3) the amount of material in motion

natural frequency

the frequency an object will vibrate when it is disturbed

resonance

a vibrating object induces a vibration of the same frequency in another object . when forced vibrations match an objects natural frequency.

Sound can be characterized by its frequency, its wavelength, its speed, and its intensity (or loundness). Sound waves carry energy that can do work (example: a sonic boom can break windows).

Want to see a picture of a real sonic boom? Sonic Boom Picture

Sources of Sound

node

region of zero displacement in a standing wave (destructive interference)

antinode

region of maximum displacement in a standing wave (constructive interference)

Sources of musical sound: Most instruments involve more than a single vibrating body. For example, in a violin, both the strings and the violin body vibrate.

• vibrating strings (guitar, piano, violin)
• vibrating membranes (drums)
• vibrating air columns (flute, oboe, organ)
• vibrating steel bars (xylophone)

5. strings produce transverse waves; sound is produced as string compresses and rarefacts air

law of strings:

• frequency is increased as string length is decreased
• frequency is increased as string diameter is decreased
• frequency is increased as string tension is increased
• frequency is increased as string density is decreased

in a standing wave on a string, each segment is ½ l

6. Pipes produce standing waves

§ closed pipes — an antinode is always at an open end and a node is always at a closed end

§ open pipes — an antinode is at each open end

Instruments produce standing waves. In any instrument, several harmonics are excited at the same time and the resultant sound is the superposition of these components.

fundamental (1^st harmonic):

o string, length = ½ l

o closed pipe, length = ¼ l

o open pipe, length = ½ l

2^nd harmonic:

o string, length = l

o open pipe, length = l

3^rd harmonic:

o string, length = 3/2 l

o closed pipe, length = ¾ l

o open pipe, length = 3/2 l

Here is a trick to remember: Draw the desired harmonic for the string, open pipe, or closed pipe. Determine how much of a wavelength is represented. Set this equal to the length of the pipe and solve for the wavelength. In the pictures above of the harmonics, if it looks like a "v" it is equal to 1/4 l. If it looks like two "v's" stuck together to form a closed object (a segment), it is equal to 1/2 l.

Notice: There are no even-numbered harmonics in a closed pipe. A closed pipe only produces odd harmonics. In strings and open pipes,

f=(n v)/2 l, where n=1, 2, 3, ... In closed pipes,

f=(n v)/4 l, where n=1, 3, 5, ...

Where l is the length of the pipe.

In music, harmonics are called overtones.

Beats Suppose two sounds with frequencies very close to one another are played simultaneously. We hear an average of the two sounds. The sound is modulated by a slow, wobbling beat note whose frequency is the difference between the two sound frequencies, or beats. For example, when a 552 Hz and a 564 Hz tone are played simultaneously, we hear 564-552, or 12 beats per second. The beat frequency is 12 Hz.

Beats - you actually HEAR them!

Interference of Sound Waves

Two speakers which emit identical sinusoidal waves of identical frequencies are another example of sound wave interference phenomena. Suppose the speakers are separated by distance d.

A microphone is placed equidistant from both speakers, on a line perpendicular to the line connecting the speakers as shown below.

Wave crests emitted from the two speakers travel equal distances to arrive at the microphone and thus arrive at the microphone at the same time. According to the principle of superposition, the amplitudes of the two waves add, resulting in constructive interference. If the microphone is moved to another position, destructive interference occurs where the wave from one speaker travels a half-wavelength farther than the wave from the other speaker. According to superposition, the amplitudes of the two waves subtract.

• You might be asked to calculate the minimum frequency where destructive interference can occur. Remember - destructive interference every 1/2 wavelength. Thus, the minimum frequency would occur when d = 1/2 l. Knowing v=lf, the speed of sound and d can be used to calculate this minimum frequency.
• You might be asked to graph how intensity varies with horizontal distance. Remember, intensity follows an inverse square relationship.
• You might be asked to graph how intensity varies with vertical distance. Remember, this looks like double slit diffraction pattern. At the midpoint, the intensity is the greatest. As you move outwards vertically, a minimum next occurs. As you continue to move out vertically, another maximum occurs, but it will not be as intense as the first one. This is followed by another minimum and so forth.