The linear density of a string is 1.6 ×10^-4 kg / m . A transverse wave on the string is descri

step 1 So in this question, we're given this formula for a transverse wave, and we're told that the mas

The linear density of a string is 1.6 ×10^-4 kg / m . A...

The linear density of a string is $1.6 \times 10^{-4} \mathrm{~kg} / \mathrm{m} .$ A transverse wave on the string is described by the equation $$ y=(0.021 \mathrm{~m}) \sin \left[\left(2.0 \mathrm{~m}^{-1}\right) x+\left(30 \mathrm{~s}^{-1}\right) t\right] $$ What are (a) the wave speed and (b) the tension in the string?

The linear density of a string is $1.6 \times 10^{-4} \mathrm{~kg} / \mathrm{m} .$ A transverse wave on the string is described by the equation
$$
y=(0.021 \mathrm{~m}) \sin \left[\left(2.0 \mathrm{~m}^{-1}\right) x+\left(30 \mathrm{~s}^{-1}\right) t\right]
$$
What are (a) the wave speed and (b) the tension in the string?

Key Concepts

Linear Density

Linear density, denoted by ?, is the mass per unit length of a medium, such as a string. It plays a crucial role in wave propagation because it influences the inertial properties of the medium. A higher linear density typically results in slower wave speeds for a given restoring tension.

Tension in a String

The tension in a string provides the restoring force necessary for transverse wave propagation. The relationship between wave speed (v), tension (T), and linear density (?) is given by v = ?(T/?). This equation allows one to calculate the tension in the string if the wave speed and linear density are known, highlighting the interplay between mechanical properties and wave motion.

Wave Equation

A wave equation, such as y(x, t) = A sin(kx ± ?t), describes the displacement of a point on a medium as a function of position and time. This equation specifies the amplitude (A), the wave number (k), which represents the spatial frequency, and the angular frequency (?), which represents the temporal frequency. It forms the basis for understanding the behavior and propagation of waves.

Wave Speed

The wave speed is the rate at which a wave propagates through a medium. It is given by the ratio of the angular frequency (?) to the wave number (k), expressed as v = ?/k. This relationship connects the characteristics of the wave’s oscillation in time and space, allowing the determination of how fast the wave fronts move.

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