Question

: A wave on a string has the formula y = 0.6sin (0.5x – 2.5t + π/2). Find the wavelength, frequency, period, phase constant, and speed of the wave? The string has a linear density μ = 0.2 kg/m. What is the tension in the string?

Answer 1

Given:

$y = 0.6 \sin(0.5x - 2.5t + \dfrac{\pi}{2} )$

To find:

• Wavelength
• Frequency
• Phase constant
• Speed of wave

Calculation:

The general equation of any progressive wave is:

$\boxed{ \bf y = a \sin( kx - \omega t + \theta) }$

Comparing with this equation, we can say:

• k = 0.5
• $\omega$ = 2.5
• $\theta$ = π/2
• a = 0.6

So, wavelength is :

$\lambda = \dfrac{2\pi}{k} = \dfrac{2\pi}{0.5} = 4\pi \: m$

So, frequency is :

$f = \dfrac{ \omega}{2\pi} = \dfrac{2.5}{2\pi} = 0.39 \: hz$

So, phase constant is :

$\theta = \dfrac{\pi}{2}$

So, wave velocity (or speed) is :

$v = \lambda \times f$

$\implies \: v = 4\pi \times 0.39$

$\implies \: v \approx 5 \: m {s}^{ - 1}$

So, tension in string is :

$v = \sqrt{T/\mu}$

$\implies T= \mu v^{2}$

$\implies T =0.2\times 25$

$\implies T =5 N$

Hope It Helps.