Interference
The principle of superposition: when two waves meet one another the net displacement of the medium at
a particular point equals the sum of the displacements of the waves at that point.
Interference is the behavior of two or more waves occupying the same space at the same time. When two
or more waves occupy the same location in a medium at the same time the medium responds to the waves
by moving in a manner that is the algebraic sum of each individual disturbance. The point of no disturbance
is called a node and the point where the interference is maximum constructive is called an antinode.
1. Each of the following graphs show two waves approaching each other and overlap between the two
dots on the line. Draw the interference pattern the two wave will make while they are overlapped.
Standing Waves on Strings
When a stretched string is plucked it will vibrate in its fundamental mode in a single segment with nodes on
each end and an antinode in the center. If the string is driven at this fundamental frequency, a standing wave
is formed. Standing waves also form if the string is driven at any integer multiple of the fundamental
frequency. These higher frequencies are called the harmonics.
Assume this rope is 1.8 m long and in (a) the frequency is 14 Hz, the fundamental frequency. The
wavelength would be 3.6 m, the length of 2 nodes. The speed is the frequency times the wavelength, so the
velocity in the rope is (14 Hz)(3.6 m) = 50.4 m/s.
In (b), this is called the 2nd harmonic (2 loops) or first overtone. Here there are 2 loops, so the length of the
rope is the wavelength, 1.8 m. The frequency is twice the fundamental, or 28 Hz. The speed of this wave is
(28 Hz)(1.8 m) = 50.4 m/s.
In (c), this is the 3rd harmonic (3 loops) or the 2nd overtone. With the three loops, each loop will have a
length of 0.6 m. This means the wavelength (length of 2 loops) is 1.2 m. The frequency is 3 times the
fundamental, or 42 Hz. The speed of this wave is (42 Hz)(1.2 m) = 50.4 m/s.
Harmonics are waves that are whole number multiples of the fundamental. Harmonics have nodes at the
boundaries. Harmonics sound louder, keep their energy longer, and take less energy to produce.
2. The string below is 2.5 m long and vibrates by the data give. Determine the harmonic, frequency,
wavelength, and velocity for each standing wave.
3. The following questions are about the standing wave to the right.
a) What is the period?
b) What harmonic is this?
c) What is the fundamental frequency?
d) If the string is 5 m long, what is the wavelength?
e) What is the speed of the wave?
f) What is the frequency of the 5th harmonic?
4. A string has a third harmonic of 15 Hz, find the frequency of harmonic 6?
5. If 35 Hz is the seventh harmonic, what is the fundamental frequency?
6. In the situations below, sketch the shape of the medium when the wave pulses are completely on top
of each other.
a.
c.
b.
7. To demonstrate standing waves, one end of a
string is attached to a wave generator with
frequency 115 Hz. The other end of the string
passes over a pulley and is connected to a
suspended 0.200 kg mass as shown in the
figure. The length of the string from the
tuning fork to the point where the string
touches the top of the pulley is 2.2 m. The
linear density of the string remains constant
throughout the experiment.
(a) Determine the wavelength of the standing
wave.
(b) Determine the velocity of the wave in the string if the wave generator vibrates at 115 Hz.
(c) Determine the fundamental frequency of the string.
(d) Determine the linear density of the string.
(e) If a point on the string at an antinode moves a total vertical distance of 6 cm during one complete cycle,
what is the amplitude of the standing wave?