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Question: A travelling wave on a string is given by y \( = \)A sin\(\left[ {\alpha x + \beta t + \dfrac{\pi }{...

A travelling wave on a string is given by y ==A sin[αx+βt+π6]\left[ {\alpha x + \beta t + \dfrac{\pi }{6}} \right]. The displacement and velocity of oscillation of a point α=0.56/cm,  β=12/sec⁡ A=7.5cm,x=1cm\alpha = 0.56/cm,\,\,\beta = 12/\sec \,A = 7.5cm,x = 1cm and t=1st = 1s is
A. 4.6cm,  46.5cms−14.6cm,\,\,46.5cm{s^{ - 1}}
B. 3.75cm,  77.94cms−13.75cm,\,\,77.94cm{s^{ - 1}}
C. 1.76cm,  7.5cms−11.76cm,\,\,7.5cm{s^{ - 1}}
D. 7.5cm,  75cms−17.5cm,\,\,75cm{s^{ - 1}}

Explanation

Solution

The travelling waves are the wave in which the position of maximum and minimum amplitude travel through the medium. Mathematically, A travelling wave is a periodic function of one dimensional space that moves with constant speed. The standard equation of a travelling wave is given as, y(x,t)=A sin⁡ (wt±kx+ϕ)y\left( {x,t} \right) = A\,\sin \,\left( {wt \pm kx + \phi } \right).

Complete step by step solution:
Given that α=0.56/cm,  β=12/sec⁡ A=7.5cm,x=1cm\alpha = 0.56/cm,\,\,\beta = 12/\sec \,A = 7.5cm,x = 1cmand t=1st = 1s
The displacement of travelling wave is,
y=A  sin⁡   (αx+βt+π6)......(i)y = A\,\,\sin \,\,\,\left( {\alpha x + \beta t + \dfrac{\pi }{6}} \right)......\left( i \right)
Putting the above values in equation (i)
y=7.5  sin⁡[(0.56)(1)+(12)×(1)+π6]y = 7.5\,\,\sin \left[ {\left( {0.56} \right)\left( 1 \right) + \left( {12} \right) \times \left( 1 \right) + \dfrac{\pi }{6}} \right]
y≃7.5×0.5y \simeq 7.5 \times 0.5
y≃3.75cmy \simeq 3.75cm

The velocity of the travelling wave v=dydtv = \dfrac{{dy}}{{dt}}
Putting the value of y from equation (i) into v=dydtv = \dfrac{{dy}}{{dt}}
So, v=dydt[A sin⁡(αx+βt+π6)]v = \dfrac{{dy}}{{dt}}\left[ {A\,\sin \left( {\alpha x + \beta t + \dfrac{\pi }{6}} \right)} \right]
⇒\Rightarrow v=Addt[sin⁡(αx+βt+π6)]v = A\dfrac{d}{{dt}}\left[ {\sin \left( {\alpha x + \beta t + \dfrac{\pi }{6}} \right)} \right]
⇒\Rightarrow v=Aβ cos⁡(αx+βt+π6)v = A\beta \,\cos \left( {\alpha x + \beta t + \dfrac{\pi }{6}} \right)
⇒\Rightarrow v=7.5×12×cos⁡[(0.56)(1)+12×1+π6]v = 7.5 \times 12 \times \cos \left[ {\left( {0.56} \right)\left( 1 \right) + 12 \times 1 + \dfrac{\pi }{6}} \right]
⇒\Rightarrow v≃7.5×12×0.8v \simeq 7.5 \times 12 \times 0.8
⇒\Rightarrow v≃77.94cm/sv \simeq 77.94cm/s

Hence, option (B) is correct.

Additional Information: If displacement equation of travelling wave is given say y then the velocity will be v=dydt.v = \dfrac{{dy}}{{dt}}.

Note: The travelling waves transport energy from one area of space to another, whereas standing waves do not transport energy.