String A is stretched between two clamps separated by...

String $A$ is stretched between two clamps separated by distance $L .$ String $B,$ with the same linear density and under the same tension as string $A,$ is stretched between two clamps separated by distance 4$L .$ Consider the first eight harmonics of string $B .$ For which of these eight harmonics of $B$ (if any) does the frequency match the frequency of (a) $A$ 'st harmonic, (b) $A$ 's second har- monic, and (c) $A$ 's third harmonic?

String $A$ is stretched between two clamps separated by distance $L .$ String $B,$ with the same linear density and under the same
tension as string $A,$ is stretched between two clamps separated by
distance 4$L .$ Consider the first eight harmonics of string $B .$ For which of these eight harmonics of $B$ (if any) does the frequency
match the frequency of (a) $A$ 'st harmonic, (b) $A$ 's second har-
monic, and (c) $A$ 's third harmonic?

Key Concepts

Harmonic Frequencies and Mode Numbers

The frequencies of the vibrational modes (or harmonics) of a string under tension are determined by the relationship f? = (n/2L)v, where 'n' is a positive integer (mode number), L is the length of the string, and v is the wave speed on the string. This formula shows that the frequency is integer-multiples of the fundamental frequency (first harmonic) and explains how overtones are related to the lowest mode.

Effect of String Length on Frequency

The length of the string plays a critical role in determining the frequency of the harmonics. Since the harmonic frequencies are inversely proportional to the string length, a string that is longer by a factor (such as 4L compared to L) will have its corresponding harmonic frequencies scaled down by that factor. This concept is key when comparing the vibrational frequencies of two strings under the same tension and linear density but different lengths.

Standing Waves on a String

When a string is fixed at both ends, it supports standing waves, which are formed by the superposition of waves reflecting from the fixed boundaries. These standing waves occur at frequencies called harmonics, where the fixed endpoints serve as nodes. Understanding the formation of standing waves is crucial because it explains why only specific vibrational modes exist in a clamped string.

Frequency Matching Among Harmonics

The idea of frequency matching involves identifying when a harmonic from one string has the same frequency as a harmonic from another string. This requires equating the frequency formulas for each harmonic of the different strings, taking into account their respective lengths. Such an analysis typically involves matching integer multiples across the two frequency systems to determine which modes coincide.

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