A transverse wave propagating along x-axis is represented by $y(x, t)=8.0 \sin(0.5\pi x-4\pi t-\pi/4)$ where x is in metres and t is in seconds. The speed of the wave is :-

Question

A steel wire with mass per unit length $ 7.0 \times 10^{-3} \, \text{kg/m} $ is under a tension of $ 70 \, \text{N} $. The speed of transverse waves in the wire will be:

Answer 1

To determine the speed of the transverse wave described by the equation $y(x, t) = 8.0 \sin(0.5\pi x - 4\pi t - \pi/4)$, we need to understand the form of a standard wave equation. A standard wave equation is of the form:

$y(x, t) = A \sin(kx - \omega t + \phi)$

Where:

A is the amplitude.
k is the wave number, given by k = \frac{2\pi}{\lambda}, where \lambda is the wavelength.
\omega is the angular frequency, given by \omega = 2\pi f, where f is the frequency.
\phi is the phase angle.

By comparing the given wave equation with the standard form, we can identify:

k = 0.5\pi
\omega = 4\pi

The speed of the wave ($v$) is given by the relationship:

v = \frac{\omega}{k}

Substituting the identified values:

v = \frac{4\pi}{0.5\pi} = \frac{4\pi}{0.5\pi} = \frac{4}{0.5} = 8

Therefore, the speed of the wave is 8 \, \text{m/s}.

Answer 2