SAT Physics Subject Test
Chapter 13 Waves
When our prototype traveling wave on a string strikes the wall, the wave will reflect and travel back toward us. The string now supports two traveling waves; the wave we generated at our end, which travels toward the wall, and the reflected wave. What we actually see on the string is the superposition of these two oppositely directed traveling waves, which have the same frequency, amplitude, and wavelength. If the length of the string is just right, the resulting pattern will oscillate vertically and remain fixed. The crests and troughs no longer travel down the length of the string. This is a standing wave, another type of wave that is important for you to know about for this test.
The right end of the string is fixed to the wall, and the left end is oscillated through a negligibly small amplitude so that we can consider both ends to be fixed (no vertical oscillation). The interference of the two traveling waves results in complete destructive interference at some points (marked N in the figure below), and complete constructive interference at other points (marked A in the figure). Other points have amplitudes between these extremes. Notice another difference between a traveling wave and a standing wave: While every point on the string had the same amplitude as the traveling wave went by, each point on a string supporting a standing wave has an individual amplitude. The points marked N are called nodes, and those marked A are called antinodes.
You can remember nodes
as areas of “no displacement.”
Antinodes are the
opposite of that.
Nodes and antinodes always alternate, they’re equally spaced, and the distance between two successive nodes (or antinodes) is equal to λ. This information can be used to determine how standing waves can be generated. The following figures show the three simplest standing waves that our string can support. The first standing wave has one antinode, the second has two, and the third has three. The length of the string in all three diagrams is L.
For the first standing wave, notice that L is equal to 1(λ). For the second standing wave, L is equal to 2(λ), and for the third, L = 3(λ). A pattern is established: A standing wave can only form when the length of the string is a multiple of λ.
Solving this for the wavelength, we get
These are called the harmonic (or resonant) wavelengths, and the integer n is known as the harmonic number.
Since we typically have control over the frequency of the waves we create, it’s more helpful to figure out the frequencies that generate a standing wave. Because λf = v, and because v is fixed by the physical characteristics of the string, the special λ’s found above correspond to equally special frequencies. From fn = v/λn, we get
These are the harmonic (or resonant) frequencies. A standing wave will form on a string if we create a traveling wave whose frequency is the same as a resonant frequency. The first standing wave, the one for which the harmonic number, n, is 1, is called the fundamental standing wave. From the equation for the harmonic frequencies, we see that the nth harmonic frequency is simply n times the fundamental frequency.
fn = nf1
Likewise, the nth harmonic wavelength is equal to λ1 divided by n. Therefore, if we know the fundamental frequency (or wavelength), we can determine all the other resonant frequencies and wavelengths.
7. A string of length 12 m that’s fixed at both ends supports a standing wave with a total of 5 nodes. What are the harmonic number and wavelength of this standing wave?
Here’s How to Crack It
First, draw a picture.
This shows that the length of the string is equal to 4(λ), so
This is the fourth-harmonic standing wave, with wavelength λ4 (because the expression above matches λn = 2L/n for n = 4). Since L = 12 m, the wavelength is
A piano tuner causes a piano string to vibrate. He then loosens the string a little, decreasing its tension, without changing the length of the string.
8. What happens to the fundamental frequency?
9. What happens to the fundamental wavelength?
Here’s How to Crack It
8. The fundamental frequency for a standing wave is given by the equation f1 = where v is the speed of a wave on the string and L is the length. The length L doesn’t change here, so f1 depends only on how v changes. We know that the speed of a wave along a string decreases as the tension decreases (look back at the equation preceding example); that is, v will decrease. Therefore, f1 will decrease, too.
9. The fundamental wavelength is given by λ1 = 2L. Since L doesn’t change, neither will λ1.