Waves
Physics Club 2018-2019
Denis Hu
1
Definition
A periodic disturbance that carries energy through matter or space, with little or no associated
mass transport. Consists of vibrations or oscillations of a physical medium or a field.
1.1
Types of Waves
Mechanical: propagate through a physical matter whose substance is being deformed; restoring
forces can reverse this deformation.
Electromagnetic: do not require a medium to travel; consists of electric and magnetic fields oscillating together.
Transverse: oscillations perpendicular to the propagation of energy transfer.
Longitudinal: oscillations parallel to the propagation of energy transfer.
1.2
Qualities of Waves
Amplitude (A): the maximum displacement of the particles in a wave from their equilibrium position.
Wavelength (λ): the length between consecutive wave crests or troughs.
Frequency (f ): the number of wavelengths that travel past a specific point in 1 second.
Speed (v): how quickly one particle in the wave travels; varies for different types of waves.
1.3
The Wave Equation
y = sin(kx − ωt)
2
2.1
Application
Doppler Effect
The Doppler effect is the change in frequency or wavelength of a wave in relation to an observer
who is moving relative to the wave source. This changes the way the observer perceives the wave.
d
The changed frequency can be calculated by fd = fs v−v
v−vs , where fd is the detected frequency,
fs the original frequency, vd is velocity of the detector, and vs is the velocity of the wave’s source.
Most problems regarding the Doppler effect involve sound; keep in mind that the speed of sound
changes with time: vsound = 331 + 0.6T , where T is the temperature in degrees Celsius.
2.2
Music and Sound
Many woodwind instruments use a concept called resonance. They have mouthpieces that alter the
sound of vibrating lips, which is what people do to play these instruments. The vibration of lips
alone is not enough; the air within the mouthpiece and the instrument tube vibrates at the same
frequency, or in resonance, with the player’s lips to create this music. For many instruments such
as flutes, changing the length of the tube will change the pitch of the instrument.
Resonance can also occur with a vibrating tuning fork above an air column. In a closed-end
air column, a wave of high and low pressure travels down. When it hits the end, it is reflected
back up to the tuning fork. If the reflected high-pressure wave reaches the tuning fork at the exact
moment it produces another high-pressure wave, the emitting and returning waves will reinforce
each other. This reinforcement produces a standing wave, and thus resonance is achieved.
In a standing wave, there are nodes and antinodes. Nodes are points along a wave where amplitude
is at a minimum., and antinodes are points where amplitude is at a maximum. The shortest air
column that can have an antinode at its closed end and a node at its second end is 14 of a wavelength long. As the frequency increases, additional resonance lengths are found at half-wavelength
5λ
intervals. Thus, columns of length λ4 , 3λ
4 , 4 , etc. will all be in resonance with the tuning fork.
Similarly, in an open pipe, the only lengths that produce nodes at both ends are found at halfwavelength intervals, starting with λ2 .
A beat is an interference pattern between two slightly different frequencies. It is perceived as
a periodic variable in volume whose rate is the difference of the two frequencies; fbeat = |fa − fb |.
3
Example
A wave has a frequency of 500Hz and travels at 350m/s. How far apart are two points that differ
in phase by π3 rad?
2
Using v = f λ, the length of one cycle of the wave is 700mm. Using f =
cycle of oscillation is 2ms.
1
T,
the time for one
One cycle is equivalent to 2π radians, so f racπ3 corresponds to one sixth of a cycle. The corresponding length, therefore, is λ6 , or 700
6 = 117mm.
4
Problems
1. A fire truck is moving at 35m/s, and a car in front of the truck is moving in the same direction
at 15m/s. If a 327Hz siren blares from the truck, what is the frequency heard from the car?
2. A flute acts as an open pipe. If the flute sounds a note with a 370Hz pitch, what are the
frequencies of the second, third, and fourth harmonics of this pitch?
3. The equation of a transverse wave travelling along a very long string is y = 6.0sin(0.020+4.0),
where x and y are in centimeters and t is in seconds. Determine the amplitude, wavelength,
frequency, and speed of the wave.
4. On a day where the temperature is 15°C, a person stands some distance from a cliff and claps
his hands. The echo returns in 2s. How far away is the cliff?
5. A sinusoidal wave travels along a strong. The time for a particular point to move from
maximum displacement to zero is 0.170s. What is the period and the frequency of the wave?
3
5
Hints
1. Use the Doppler formula.
2. What is the wavelength that each harmonic corresponds to?
3. Use the given equation.
4. The speed of sound changes with temperature.
5. Maximum displacement to zero is one fourth of a cycle.
4