Vibrations with frequency `6.00 xx 10^2` Hz are established on a 1.33 m length of string that is clamed at both ends. The speed of waves on the string is `4.0 xx 10^2 m//s`
How many antinodes are contained in the resulting standing wave pattern?

Options

A. 2
B. 4
C. 3
D. 5

Answer

To find the number of antinodes in the standing wave pattern formed on a string clamped at both ends, we can follow these steps:

### Step 1: Understand the relationship between harmonics and standing waves
For a string clamped at both ends, the standing wave pattern consists of nodes and antinodes. The number of antinodes is related to the harmonic of the wave. The nth harmonic has n antinodes and (n + 1) nodes.

### Step 2: Use the formula for the wavelength in terms of the harmonic
The relationship between the length of the string (L), the wavelength (λ), and the harmonic number (n) is given by:
\[ L = \frac{n \lambda}{2} \]
This means that the length of the string is equal to n half-wavelengths.

### Step 3: Relate wavelength to wave speed and frequency
The wavelength can also be expressed in terms of wave speed (v) and frequency (f):
\[ \lambda = \frac{v}{f} \]

### Step 4: Substitute the expression for wavelength into the length equation
Substituting the expression for λ into the length equation gives:
\[ L = \frac{n}{2} \cdot \frac{v}{f} \]

### Step 5: Rearrange to find n
Rearranging the equation to solve for n gives:
\[ n = \frac{2Lf}{v} \]

### Step 6: Substitute the known values
Given:
- Length of the string, \( L = 1.33 \, m \)
- Frequency, \( f = 6.00 \times 10^2 \, Hz = 600 \, Hz \)
- Wave speed, \( v = 4.0 \times 10^2 \, m/s = 400 \, m/s \)

Now, substituting these values into the equation:
\[ n = \frac{2 \times 1.33 \times 600}{400} \]

### Step 7: Calculate n
Calculating the above expression:
\[ n = \frac{2 \times 1.33 \times 600}{400} = \frac{1596}{400} = 3.99 \approx 4 \]

### Step 8: Determine the number of antinodes
Since we found \( n = 4 \), the number of antinodes in the standing wave pattern is equal to \( n \):
\[ \text{Number of antinodes} = n = 4 \]

### Final Answer
Thus, the number of antinodes contained in the resulting standing wave pattern is **4**.

---