A. What are the three longest wavelengths for standing waves...

a. What are the three longest wavelengths for standing waves on a $240-\mathrm{cm}$ -long string that is fixed at both ends?
b. If the frequency of the second-longest wavelength is $50 \mathrm{Hz}$, what is the frequency of the thind-longest wavelength?

Key Concepts

Wavelength Quantization

Due to the fixed-end boundary conditions, the allowed wavelengths are quantized and given by the relation ?_n = 2L/n, where L is the length of the string and n is the positive integer representing the harmonic number. This formula helps identify the distinct wavelengths, such as the three longest ones in a problem scenario.

Harmonics and Mode Numbers

The standing waves on a fixed string exist only at specific frequencies called harmonics. Each harmonic corresponds to a mode number (n), where the fundamental (n=1) has the longest wavelength and higher harmonics (n=2, 3, ...) have shorter wavelengths. These modes describe the pattern of nodes and antinodes on the string.

Frequency-Wave Speed Relationship

In any wave motion, the frequency, wavelength, and wave speed are related by v = f?. For standing waves on a string, once the allowed wavelengths are determined by the mode number and string length, knowing one of the frequencies (for example, from a particular harmonic) can allow calculation of the wave speed, which in turn is used to compute the frequencies of other modes.

Standing Waves

Standing waves occur when two waves of the same frequency and amplitude traveling in opposite directions interfere, creating a pattern of nodes and antinodes. This phenomenon is key in understanding resonance and vibration in systems such as strings and air columns.

Fixed-End Boundary Conditions

When a string is fixed at both ends, the ends must always be nodes, meaning the displacement there is zero. This constraint results in only certain wavelengths (and thus frequencies) being possible, which are determined by the length of the string.

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