A tube with both ends closed has same set of natural frequency as

Options

A. one end closed organ pipe
B. both end open organ pipe
C. vibratory string fixed at both ends
D. vibratory string fixed at one end

Answer

To determine the natural frequencies of a tube with both ends closed, we can follow these steps:

### Step 1: Understand the configuration of the tube
A tube that is closed at both ends will have nodes at both ends. This means that the displacement of the air particles at the ends of the tube is zero.

### Step 2: Relate the length of the tube to the wavelength
For a tube closed at both ends, the fundamental frequency (first harmonic) corresponds to a standing wave where the length of the tube (L) is equal to half the wavelength (λ/2). Therefore, we can express this relationship as:
\[ L = \frac{\lambda}{2} \]

### Step 3: Solve for the wavelength
From the equation \( L = \frac{\lambda}{2} \), we can rearrange it to find the wavelength:
\[ \lambda = 2L \]

### Step 4: Determine the frequency
The frequency (f) of a wave is related to its speed (v) and wavelength (λ) by the equation:
\[ f = \frac{v}{\lambda} \]
Substituting the expression for λ from step 3 into this equation gives:
\[ f = \frac{v}{2L} \]

### Step 5: Compare with an open tube
For a tube that is open at both ends, the fundamental frequency also results in a similar relationship. In this case, the length of the tube is equal to half the wavelength (λ/2), leading to the same expression for frequency:
\[ f = \frac{v}{2L} \]

### Conclusion
Thus, we find that a tube with both ends closed has the same natural frequency as a tube that is open at both ends, both yielding the frequency expression:
\[ f = \frac{v}{2L} \]