Step-by-step solution
First, we determine the harmonic number by looking at the wave above. Then, we calculate the wave speed using the equation for the
frequency of the nth harmonic.
Now we use the equation that relates wave speed to tension, mass, and string length.
Step Reason
1. n = 4 harmonic is one less than number of nodes
2. frequency of nth harmonic
3. solve for wave speed
4. substitute values
5. v = 105 m/s evaluate
Step Reason
6. wave speed on string
7. solve for tension
8. substitute values
9. T = 74.9 N evaluate
17.7 - Wave interference and path length
We described wave interference resulting from a phase difference, or differing initial
conditions for two waves moving in the same direction. Interference also results when
two waves travel from different starting points and meet. To illustrate this, we use the
example of two longitudinal traveling waves produced by two loudspeakers.
To the right, we show a person listening to the loudspeakers. The vibrating speakers
create regions of pressure that are greater than atmospheric pressure (condensation)
and less (rarefaction). We show this pattern of oscillation emanating from each speaker
in Concept 1. We assume the speakers create waves with the same amplitude and
wavelength, and that there is no phase difference between them. We focus on the point
where the two waves combine just as they reach the listener’s ear.
In Concept 1, we position the two speakers so that the listener is equidistant from them.
The distance from a speaker to the ear is called the path length. Since the waves travel
the same distance, they will be in phase when they arrive. This means peaks and
troughs exactly coincide with each other.
When two peaks combine, they double the pressure increase. When two troughs combine, they also add, and the pressure decrease is
doubled. The listener hears a louder sound. In short, the waves constructively interfere.
The waves would also constructively interfere if one speaker were placed one wavelength farther away. Peaks would still meet peaks and
troughs would still meet troughs. In fact, if the difference in the distances between the loudspeakers and the listener is any integer multiple of
the wavelength, the waves constructively interfere. (This does assume that the loudspeakers vibrate at a constant frequency, an unusual
assumption for most music.) You see this condition for constructive interference stated in Equation 1.
Can we arrange the speakers so that the waves destructively interfere? Yes, by moving one speaker one-half wavelength away from the
listener. This is shown in Concept 2. When the waves combine, peaks will meet troughs and vice versa. This will also occur if the
speaker/listener distances differ by half a wavelength, or 1.5 wavelengths, or any half-integer multiple of the wavelength. This condition for
destructive interference is stated as an equation on the right. If the waves have the same amplitude at the ear and destructively interfere
completely, the result is silence at the listener’s position.
Path lengths the same
Constructive interference