(II) Show that the frequency of standing waves on a cord of length ℓand linear density μ, which

step 1 In this problem, we have given the length of code which is equal to L. And we know that the freq

(II) Show that the frequency of standing waves on a cord of...

(II) Show that the frequency of standing waves on a cord of length $\ell$ and linear density $\mu,$ which is stretched to a tension $F_{T, \text { is given by }}$
$$f=\frac{n}{2 \ell} \sqrt{\frac{F_{\mathrm{T}}}{\mu}}$$
where $n$ is an integer.

Key Concepts

Standing Waves

Standing waves occur when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other, resulting in a fixed pattern of nodes (points of zero displacement) and antinodes (points of maximum displacement). This concept is fundamental in understanding the vibrational modes of a medium, such as a string, and is key to explaining how resonant frequencies arise in various systems.

Boundary Conditions

Boundary conditions refer to the constraints imposed on a vibrating system at its endpoints or boundaries, such as a fixed end where the displacement is always zero. These conditions determine the allowed patterns of vibration, leading to discrete modes and frequencies. For a string fixed at both ends, the condition that the displacement must be zero at these endpoints results in standing wave patterns that support only specific wavelengths.

Wave Speed on a String

The speed of a wave on a string is determined by the tension in the string and its mass per unit length, with the relationship given by the square root of the tension divided by the linear density. This relation is crucial because it connects the physical properties of the string to the propagation of waves along it, and it forms the basis for determining the frequencies of the normal modes.

Normal Modes

Normal modes refer to the particular standing wave patterns that a system can support without any change in shape over time, aside from oscillation in amplitude. In a string fixed at both ends, each normal mode corresponds to a specific integer multiple of the fundamental frequency. These modes are essential in understanding how energy is distributed in a vibrating system and are described by their discrete frequencies and spatial wave patterns.

Frequency Quantization

Frequency quantization is the phenomenon where only specific, discrete frequencies (often integer multiples of a fundamental frequency) are allowed in a confined system due to the imposed boundary conditions. In the context of a stretched string with fixed endpoints, this quantization results in a relationship between the integer mode number, the length of the string, the tension, and the linear density, leading to the formula for the standing wave frequencies.

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