# Standing Waves on a String Simulation

#### Quiz

Q1) A point on a standing wave that is always stationary is a

A. maximum
B. minimum
C. node
D. antinode

Q2) Standing waves are produced by the superposition of two waves with
A. the same amplitude, frequency, and direction of propagation.
B. the same amplitude and frequency, and opposite propagation directions.
C. the same amplitude and direction of propagation, but different frequencies.
D. the same amplitude, different frequencies, and opposite directions of propagation.

#### Standing Waves on a String Simulation

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Standing waves
If you tie a rope to a wall and shake the free end up and down, you will produce a wave in the rope. The wall is too rigid to shake, so the wave is reflected back along the rope to you. By shaking the rope just right, you can cause the incident (original) and reflected waves to form a standing wave. A standing wave is a wave that appears to stay in one place—it does not seem to move through the medium.
Certain parts of a standing wave remain stationary. Nodes are the stationary points on a standing wave. Interestingly enough, you could hold your fingers on either side of the rope at a node, and the rope would not touch them. Other parts of the rope would make contact with your fingers. The positions on a standing wave with the largest amplitudes are known as antinodes. Antinodes occur halfway between nodes.

FIGURE 1. The incident and reflected waves interfere to produce a standing wave. The nodes are places that remain stationary.

Standing waves are the result of interference. When two waves of equal amplitude and wavelength pass through each other in opposite directions, the waves are always out of phase at the nodes. As Figure 1 shows, the nodes are stable regions of destructive interference.

Standing waves have nodes and antinodes
Figure 2(a) shows four possible standing waves for a given string length. The points at which complete destructive interference happens are called nodes. There is no motion in the string at the nodes. But midway between two adjacent nodes, the string vibrates with the largest amplitude. These points are called antinodes.
Figure 2(b) shows the oscillation of the second case shown in Figure 2(a) during half a cycle. All points on the string oscillate vertically with the same frequency, except for the nodes, which are stationary. In this case, there are three nodes (N) and two antinodes (A), as illustrated in the figure.

Figure 2. (a) This photograph shows four possible standing waves that can exist on a given string. (b) The diagram shows the progression of the second standing wave for one-half of a cycle.

In general, the motion of an oscillating string fixed at both ends is described by the superposition of several normal modes. Exactly which normal modes are present depends on how the oscillation is started. For example, when a guitar string is plucked near its middle, the modes shown in Figure 3(b) and (d), as well as other modes not shown, are excited.

Figure 3. (a) A string of length L fixed at both ends. The normal modes of vibration form a harmonic series: (b) the fundamental, or first harmonic; (c) the second harmonic; (d) the third harmonic.

In general, we can describe the normal modes of oscillation for the string by imposing the requirements that the ends be nodes and that the nodes and antinodes be separated by one fourth of a wavelength. The first normal mode, shown in Figure 3(b), has nodes at its ends and one antinode in the middle. This is the longestwavelength mode, and this is consistent with our requirements. This first normal mode occurs when the wavelength $\lambda _{1}$ is twice the length of the string, that is, $\lambda _{1}=2L$.
The next normal mode, of wavelength $\lambda _{2}$ (see Fig. 3(c)), occurs when the wavelength equals the length of the string, that is, $\lambda _{2}=L$.
The third normal mode(see Fig. 3(d)) corresponds to the case in which $\lambda _{3}=2L/3$.

In general, the wavelengths of the various normal modes for a string of length $\ L$ fixed at both ends are

$\lambda _{n}=\frac{2L}{n}~~~~~~n=1,2,3,...~~~~~(1)$.

where the index $\ n$ refers to the nth normal mode of oscillation. These are the possible modes of oscillation for the string. The actual modes that are excited by a given pluck of the string are discussed below.

The natural frequencies associated with these modes are obtained from the relationship $\ f=v/\lambda$, where the wave speed $\ v$ is the same for all frequencies. Using Equation (1), we find that the natural frequencies $\ f_{n}$ of the normal modes are

$f_{n}= \frac {v}{\lambda_{n}}=n \frac{v}{2L}~~~~~~n=1,2,3,...~~~~~(2)$.