Question

A 2 m-long string fixed at both ends is set into vibrations in its first overtone. The wave speed on the string is $200m{s}^{-1}$ and the amplitude is 0.5 cm. Write the equation giving the displacement of different points as a function of time. Choose the X-axis along the string with the origin at one end and t = 0 at the instant when the point x = 50 cm has reached its maximum displacement.

• A. $sin\left(\pi x\right)cos\left(200\pi t\right)$
• B. $sin\left(2\pi x\right)cos\left(200\pi t\right)$
• C. $sin\left(\pi x\right)cos\left(400\pi t\right)$
• D. $sin\left(2\pi x\right)cos\left(400\pi t\right)$

Answer 1

The correct option is A

$sin\left(\pi x\right)cos\left(200\pi t\right)$

1st overtone refers to the 2nd harmonic :$f=2\times \frac{v}{2L}=\frac{v}{L}=\frac{200}{2}=100Hz$

Wavelength, $\lambda =\frac{v}{f}=\frac{200}{100}=2m$

Now to find the equation, we know the standard equation of a standing wave

$y=2Asin\left(kx\right)cos\left(\omega t\right)$
Hence, $y=sin\left(\pi x\right)cos\left(200\pi t\right)$