Question 83: Waves

Question

The equation of a plane progressive wave is given by $y = 5 \cos \pi \left( 200t - \frac{x}{150} \right)$ where $x$ and $y$ are in cm and $t$ is in second. The velocity of the wave is ________ m/s.

Accepted Answer

For a closed organ pipe
, only odd harmonics are present. The frequencies are:
$$f_n = \frac{nv}{4L_c}, \quad n = 1, 3, 5, 7, ...$$
The
fifth harmonic
(n = 5) of the closed pipe:
$$f_{closed} = \frac{5v}{4L_c}$$
For an open organ pipe
, all harmonics are present. The frequencies are:
$$f_n = \frac{nv}{2L_o}, \quad n = 1, 2, 3, ...$$
The
first harmonic
(n = 1) of the open pipe:
$$f_{open} = \frac{v}{2L_o}$$
Setting Frequencies Equal (Unison)
Since they are in unison:
$$\frac{5v}{4L_c} = \frac{v}{2L_o}$$
Solving for the Ratio
Cancelling $v$ from both sides:
$$\frac{5}{4L_c} = \frac{1}{2L_o}$$
Cross-multiplying:
$$5 	imes 2L_o = 4L_c 	imes 1$$
$$10L_o = 4L_c$$
$$\frac{L_c}{L_o} = \frac{10}{4} = \frac{5}{2}$$
Finding x
Given that the ratio is $\frac{5}{x}$:
$$\frac{5}{x} = \frac{5}{2}$$
$$\boxed{x = 2}$$
The answer is
Option B: 2
.

Answer

120

Answer

150

Answer

200

Answer

300

Waves

The equation of a plane progressive wave is given by y = 5 \cos \pi \left( 200t - \frac{x}{150} \right) where x and y are in cm and t is in second. The velocity of the wave is ________ m/s.

Choose the correct answer: