A travelling harmonic wave is represented by the equation y(x, t) = 10

1.	A travelling harmonic wave is represented by the equation y(x, t) = 10-3 sin (50t + 2x), where ‘x’ and ‘y’ are in metre and ‘t’ is in second. Which of the following is a correct statement about the wave?1. The wave is propagating along the negative X-axis with speed 25 m/s.2. The wave is propagating along the positive X-axis with speed 25 m/s.3. The wave is propagating along the positive X-axis with speed 100 m/s.4. The wave is propagating along the negative X-axis with speed 100 m/s.
Answer» Correct Answer - Option 1 : The wave is propagating along the negative X-axis with speed 25 m/s. Concept: From question, the wave equation given is: Y = 10^-3 sin (50t + 2x) Speed of wave is obtained by differentiating phase of wave. Calculation: Now, phase of wave from given equation is: ϕ = 50t + 2x = constant On differentiation ϕ with respect to ‘t’, \(\Rightarrow \frac{d}{{dt}}\left( {50t + 2x} \right) = \frac{d}{{dt}}{\rm{\;(constant)}}\) \(\Rightarrow 50 + 2\left( {\frac{{dx}}{{dt}}} \right) = 0\) \(\Rightarrow \frac{{dx}}{{dt}} = \frac{{ - 50}}{2}\) \(\therefore \frac{{dx}}{{dt}} = - 25{\rm{\;m}}/{\rm{s}}\) So, wave is propagating in negative x-direction with a speed of 25 m/s.

A travelling harmonic wave is represented by the equation y(x, t) = 10-3 sin (50t + 2x), where ‘x’ and ‘y’ are in metre and ‘t’ is in second. Which of the following is a correct statement about the wave?1. The wave is propagating along the negative X-axis with speed 25 m/s.2. The wave is propagating along the positive X-axis with speed 25 m/s.3. The wave is propagating along the positive X-axis with speed 100 m/s.4. The wave is propagating along the negative X-axis with speed 100 m/s.

Answer» Correct Answer - Option 1 : The wave is propagating along the negative X-axis with speed 25 m/s.

Concept:

From question, the wave equation given is:

Y = 10^-3 sin (50t + 2x)

Speed of wave is obtained by differentiating phase of wave.

Calculation:

Now, phase of wave from given equation is:

ϕ = 50t + 2x = constant

On differentiation ϕ with respect to ‘t’,

\(\Rightarrow \frac{d}{{dt}}\left( {50t + 2x} \right) = \frac{d}{{dt}}{\rm{\;(constant)}}\)

\(\Rightarrow 50 + 2\left( {\frac{{dx}}{{dt}}} \right) = 0\)

\(\Rightarrow \frac{{dx}}{{dt}} = \frac{{ - 50}}{2}\)

\(\therefore \frac{{dx}}{{dt}} = - 25{\rm{\;m}}/{\rm{s}}\)

So, wave is propagating in negative x-direction with a speed of 25 m/s.