Question

The equation of a progressive wave is \(Y = 3sin[π(\frac{t}{3}-\frac{x}{5})+\frac{π}{4}]\) where X and Y are in metre and time in second. Which of the following is correct?

• A. Velocity = \(1.5\) m/s
• B. Amplitude = \(30\) cm
• C. Wavelength = \(10\) m
• D. Frequency = \(0.2\) Hz

Hint:

Compare with Y = A sin(omega t - k x + phi) and find omega, k, wavelength and speed.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Compare the given wave with the standard form \(Y=A\sin(\omega t-kx+\phi)\).

Step 2: Expand the equation:
\(Y=3\sin\left(\dfrac{\pi}{3}t-\dfrac\pi5x+\dfrac\pi4\right)\). So \(A=3\) m, \(\omega=\dfrac\pi3\) rad/s and \(k=\dfrac\pi5\) rad/m.

Step 3: Find each quantity:
Wavelength \(\lambda=\dfrac{2\pi}{k}=\dfrac{2\pi}{\pi/5}=10\) m. Frequency \(f=\dfrac{\omega}{2\pi}=\dfrac16\) Hz. Speed \(v=\dfrac\omega k=\dfrac{\pi/3}{\pi/5}=\dfrac53\) m/s.

Step 4: Check the options:
(A) speed 1.5 m/s is wrong, it is 1.67 m/s. (B) amplitude 30 cm is wrong, it is 3 m = 300 cm. (C) wavelength 10 m is correct. (D) frequency 0.2 Hz is wrong, it is 0.167 Hz. Option C.

Final Answer:
The wavelength is 10 m. \[ \boxed{\text{(C) }\text{Wavelength}=10\ \text{m}} \]