Let's break down the problem step by step: **Given:** - Frequency, \( f = 2 \) kHz \( = 2000 \) Hz - Temperature of air = 23°C (sound velocity given as 330 m/s) - Observed differential pressure at \( x = 0, t = 70 \mu s \) is \( P(0, 70 \mu s) = 10 \, N/m^2 \) - Reference phase \( \phi_0 = 36^\circ \) - Sound velocity, \( v = 330 \) m/s We are asked to **find the complete expression for the pressure wave \( P(x, t) \).** --- ### 1. **General Form of the Sound Wave** The pressure variation for a sound wave traveling in the \( x \)-direction can be written: \[ P(x, t) = P_0 \cos (kx - \omega t + \phi_0) \] Where: - \( P_0 \) = pressure amplitude (to be found) - \( k \) = wave number \( = \frac{2\pi}{\lambda} \) - \( \omega \) = angular frequency \( = 2\pi f \) - \( \phi_0 \) = phase reference --- ### 2. **Find \( k \) and \( \omega \):** #### **Wavelength \( \lambda \):** \[ \lambda = \frac{v}{f} = \frac{330 \, \text{m/s}}{2000 \, \text{Hz}} = 0.165 \, \text{m} \] #### **Wave number \( k \):** \[ k = \frac{2\pi}{\lambda} = \frac{2\pi}{0.165} \approx 38.1 \, \text{rad/m} \] #### **Angular frequency \( \omega \):** \[ \omega = 2\pi f = 2\pi \times 2000 = 4000\pi \approx 12566.37 \, \text{rad/s} \] --- ### 3. **Plug in the values at \( x = 0, t = 70 \mu s \):** Given: - \( x = 0 \) - \( t = 70 \mu s = 70 \times 10^{-6} \) s - \( P(0, 70 \mu s) = 10 \, N/m^2 \) - Reference phase \( \phi_0 = 36^\circ \) (convert to radians: \( \phi_0 = 36^\circ \times \frac{\pi}{180} = 0.628 \) rad) The expression at \( x = 0 \): \[ P(0, t) = P_0 \cos(-\omega t + \phi_0) \] At \( t = 70 \mu s \): \[ 10 = P_0 \cos(-\omega t + \phi_0) \] Calculate \( -\omega t + \phi_0 \): \[ \omega t = 12566.37 \times 70 \times 10^{-6} = 0.8796 \, \text{rad} \] \[ -\omega t + \phi_0 = -0.8796 + 0.628 = -0.2516 \, \text{rad} \] \[ \cos(-0.2516) = \cos(0.2516) = 0.9686 \] Thus, \[ 10 = P_0 \times 0.9686 \implies P_0 = \frac{10}{0.9686} = 10.32 \, N/m^2 \] --- ### 4. **Write the Complete Expression:** Now, substitute \( P_0 \), \( k \), \( \omega \), and \( \phi_0 \): \[ \boxed{ P(x, t) = 10.32 \cos(38.1 x - 12566.4 t + 0.628) \quad \text{(with \( x \) in meters, \( t \) in seconds)} } \] --- ## **Final Answer** \[ \boxed{ P(x, t) = 10.32 \cos\big(38.1\, x - 12566.4\, t + 0.628\big) \quad \text{(N/m}^2\text{, with \( x \) in m, \( t \) in s)} } \] --- **Summary of steps:** 1. Wrote the general wave equation. 2. Calculated \( \lambda, k, \omega \). 3. Used the observed value to solve for amplitude \( P_0 \). 4. Plugged in all values to get the final, complete expression.