A transverse wave described by y=(0 ·02 m) sin[(1 ·0 m^-1) x+(30 s^-1) t] propagates on a str

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A transverse wave described by y=(0 ·02 m) sin[(1 ·0 m^-1)...

A transverse wave described by $$ y=(0 \cdot 02 \mathrm{~m}) \sin \left[\left(1 \cdot 0 \mathrm{~m}^{-1}\right) x+\left(30 \mathrm{~s}^{-1}\right) t\right] $$ propagates on a stretched string having a linear mass density of $1 \cdot 2 \times 10^{-4} \mathrm{~kg} \mathrm{~m}^{-1}$. Find the tension in the string.

A transverse wave described by
$$
y=(0 \cdot 02 \mathrm{~m}) \sin \left[\left(1 \cdot 0 \mathrm{~m}^{-1}\right) x+\left(30 \mathrm{~s}^{-1}\right) t\right]
$$
propagates on a stretched string having a linear mass density of $1 \cdot 2 \times 10^{-4} \mathrm{~kg} \mathrm{~m}^{-1}$. Find the tension in the string.

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Key Concepts

Wave Equation

The wave equation mathematically describes how displacement in a medium varies with both position and time, often represented in the sinusoidal form y(x, t) = A sin(kx ± ?t). It embodies the fundamental properties of waves such as amplitude, wave number, and angular frequency, illustrating how waves propagate through a medium.

Phase Velocity

Phase velocity is the speed at which a specific phase of the wave (for example, the crest) travels through the medium. It is determined by the ratio of the angular frequency (?) to the wave number (k), given by v = ?/k. This concept is crucial in understanding the propagation characteristics of waves.

Linear Mass Density

Linear mass density, symbolized by ?, represents the mass per unit length of a medium such as a string or wire. This inherent property of the medium affects how easily the wave can propagate, with lighter (smaller ?) strings typically allowing faster wave speeds under the same tension.

Tension and Wave Speed

In a stretched string, the tension (T) contributes directly to the wave speed. The relationship between wave speed, tension, and linear mass density is given by v = ?(T/?). This connection highlights how the physical properties of the string and the applied tension determine the behavior and speed of transverse waves.

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