Waves — NEET UG practice

1. Recall that in an open pipe (open at both ends), every harmonic — 1st, 2nd, 3rd, and so on — is allowed, and the number of nodes formed always equals the harmonic number itself.

Given the pipe supports the 5th harmonic,

$$n = 5$$

2. Recall that in a pipe closed at one end, only odd harmonics (1st, 3rd, 5th, 7th, 9th, ...) can exist, and each successive odd harmonic adds one more node than the previous one.

Counting nodes for successive odd harmonics: 1st harmonic → 1 node, 3rd harmonic → 2 nodes, 5th harmonic → 3 nodes, 7th harmonic → 4 nodes, 9th harmonic → 5 nodes.

Therefore, for the 9th harmonic,

$$m = 5$$

3. Since both $n$ and $m$ are now known, find their ratio.

Hence,

$$\frac{n}{m} = \frac{5}{5} = 1$$

Hence, the answer is D.

1. Compare the given equation with the standard form of a travelling wave to identify the wave number.

The general travelling wave equation is written as

$$y = A\cos(\omega t - kx + \phi_0)$$

Comparing with the given equation, the wave number is

$$k = 2\pi \times 0.0080 \text{ rad/cm}$$

2. Recall that the phase difference between two points separated by a distance $\Delta x$ is given by $\Delta\phi = k\Delta x$. Since $k$ is in per-cm units, first convert the given separation to cm.

Given $\Delta x = 0.5\text{ m} = 50\text{ cm}$,

$$\Delta\phi = k\Delta x$$

3. Substitute the known values into the phase difference formula.

$$\Delta\phi = 2\pi \times 0.0080 \times 50$$

4. Simplify to get the final phase difference.

$$\Delta\phi = 0.8\pi \text{ rad}$$

Hence, the answer is B.

1. Recall the fundamental frequency formula for a pipe open at both ends of length $L$, where $v$ is the speed of sound.

$$f = \frac{v}{2L}$$

2. When the pipe is dipped vertically into water up to half its length, the submerged end gets blocked by water, and the air column above the water (of length $L/2$) now behaves as a pipe closed at one end.

For a closed pipe of length $L' = L/2$, the fundamental frequency is

$$f' = \frac{v}{4L'}$$

3. Substitute $L' = L/2$ into the closed pipe formula.

$$f' = \frac{v}{4(L/2)} = \frac{v}{2L}$$

4. Compare this result with the original fundamental frequency $f = \frac{v}{2L}$ from step 1.

Since both expressions are identical,

$$f' = f$$

Hence, the answer is D.