Write the equation for the fundamental standing sound waves in a tube that is open at both ends. If the tube is `80 cm` long speed of wave is `330 m//s` . Represent the amplitude of the wave at an antinode by `A` .

Explanation

To find the equation for the fundamental standing sound waves in a tube that is open at both ends, we will follow these steps:

### Step 1: Understand the conditions for a tube open at both ends
In a tube that is open at both ends, the fundamental frequency (first harmonic) has antinodes at both ends. The length of the tube corresponds to half the wavelength of the sound wave.

### Step 2: Calculate the wavelength
The relationship between the length of the tube (L) and the wavelength (λ) for the fundamental mode is given by:
\[ L = \frac{\lambda}{2} \]
From this, we can express the wavelength as:
\[ \lambda = 2L \]
Given that the length of the tube \( L = 80 \, \text{cm} = 0.8 \, \text{m} \), we can calculate:
\[ \lambda = 2 \times 0.8 \, \text{m} = 1.6 \, \text{m} \]

### Step 3: Calculate the fundamental frequency
The fundamental frequency (f) can be calculated using the formula:
\[ f = \frac{v}{\lambda} \]
where \( v \) is the speed of sound. Given \( v = 330 \, \text{m/s} \):
\[ f = \frac{330 \, \text{m/s}}{1.6 \, \text{m}} \approx 206.25 \, \text{Hz} \]

### Step 4: Write the equation for the standing wave
The general equation for a standing wave can be expressed as:
\[ S(x, t) = A \sin(kx) \cos(\omega t) \]
where:
- \( A \) is the amplitude of the wave,
- \( k \) is the wave number, given by \( k = \frac{2\pi}{\lambda} \),
- \( \omega \) is the angular frequency, given by \( \omega = 2\pi f \).

Calculating \( k \):
\[ k = \frac{2\pi}{1.6} \approx 3.93 \, \text{m}^{-1} \]

Calculating \( \omega \):
\[ \omega = 2\pi \times 206.25 \approx 1292.5 \, \text{s}^{-1} \]

### Final Equation
Substituting \( A \), \( k \), and \( \omega \) into the standing wave equation, we get:
\[ S(x, t) = A \sin(3.93 x) \cos(1292.5 t) \]

### Summary
The equation for the fundamental standing sound wave in a tube that is open at both ends, with a length of 80 cm and a wave speed of 330 m/s, is:
\[ S(x, t) = A \sin(3.93 x) \cos(1292.5 t) \]