The fundamental frequency of a string is proportional to

Options

A. inverse of its length
B. the diameter
C. the tension
D. the density

Answer

To determine what the fundamental frequency of a string is proportional to, we need to analyze the relationship between frequency, wavelength, and the physical properties of the string.

### Step-by-Step Solution:

1. **Understanding the Fundamental Frequency**:
- The fundamental frequency (first harmonic) of a string fixed at both ends can be expressed in terms of its length (L) and wavelength (λ). For a string fixed at both ends, the relationship is:
\[ L = \frac{\lambda}{2} \]
- This implies that the wavelength for the fundamental frequency is:
\[ \lambda = 2L \]

2. **Frequency Formula**:
- The frequency (f) of a wave is given by the formula:
\[ f = \frac{v}{\lambda} \]
- Where v is the speed of the wave on the string.

3. **Substituting Wavelength**:
- For the fundamental frequency of a string fixed at both ends, substituting the wavelength we found:
\[ f = \frac{v}{2L} \]
- This shows that the fundamental frequency is inversely proportional to the length of the string (L).

4. **Considering a String with One End Fixed**:
- For a string that is fixed at one end and free at the other, the relationship changes:
\[ L = \frac{\lambda}{4} \]
- Thus, the wavelength for this case is:
\[ \lambda = 4L \]
- The frequency for this case can be expressed as:
\[ f = \frac{v}{\lambda} = \frac{v}{4L} \]
- Again, we see that the fundamental frequency is inversely proportional to the length of the string.

5. **Conclusion**:
- From both cases, we conclude that the fundamental frequency of a string is inversely proportional to its length (L). Therefore, we can summarize:
\[ f \propto \frac{1}{L} \]

### Final Answer:
The fundamental frequency of a string is inversely proportional to its length (L).