# 1. Pressure vs. Position at Fixed Time (Sound Wave Left) We consider a **sound wave traveling to the left** along the \( x \)-axis. The over-pressure (pressure minus atmospheric pressure) as a function of position at a fixed instant is a **cosine or sine wave**, depending on phase. ## Step-by-Step Solution ### a. Upper Graph: Pressure vs. Position at Time \( t_1 \) - **Horizontal Axis:** Position (\( x \)) - **Vertical Axis:** Over-pressure (\( p(x, t_1) \)) - For a wave traveling left: \[ p(x, t) = A \cos(kx + \omega t + \phi_) \] At fixed \( t_1 \), plot as a cosine with respect to \( x \). - **Arrow:** Mark one full wavelength (\( \lambda \)) along the \( x \)-axis. **Sketch:** ``` | /‾‾‾‾\ /‾‾‾‾\ (vertical: p, horizontal: x) | / \ / \ |------/ \__/ \_____ <------λ------> ``` *Alt text: A sinusoidal curve versus position with a horizontal arrow indicating one wavelength.* ### b. Lower Graph: Pressure vs. Position at Later Time \( t_2 > t_1 \) - At a slightly later time, the whole wave has shifted **to the left** (since it's moving left). - The sinusoidal pattern repeats, shifted left by a distance \( v \Delta t \) (where \( v \) is wave speed and \( \Delta t = t_2 - t_1 \)). **Sketch:** ``` | /‾‾‾‾\ /‾‾‾‾\ (vertical: p, horizontal: x) | / \ / \ |/ \__/ \_____ <------λ------> ``` *Alt text: The same sinusoidal over-pressure curve shifted left along the position axis, with a horizontal arrow showing one wavelength.* ### **Labels** - **Vertical axis:** Over-pressure, \( p(x, t) \) - **Horizontal axis:** Position, \( x \) - **Arrow:** One wavelength, \( \lambda \) --- # 2. Pressure vs. Time at Fixed Position (Same Wave) Now, consider **pressure as a function of time at a fixed point** as the wave passes by. ## Step-by-Step Solution ### a. Upper Graph: Pressure vs. Time at \( x_1 \) - **Horizontal Axis:** Time (\( t \)) - **Vertical Axis:** Over-pressure (\( p(x_1, t) \)) - The function: \[ p(x_1, t) = A \cos(kx_1 + \omega t + \phi_) \] At fixed \( x_1 \), plot as cosine in time. - **Arrow:** Mark one period (\( T \)) on the time axis. **Sketch:** ``` | /‾‾‾‾\ /‾‾‾‾\ (vertical: p, horizontal: t) | / \ / \ |_____/ \__/ \_____ <------T------> ``` *Alt text: Sinusoidal over-pressure curve versus time with a horizontal arrow indicating one period.* ### b. Lower Graph: Pressure vs. Time at \( x_2 > x_1 \) - At a position **slightly to the right** (\( x_2 > x_1 \)), the wave arrives **slightly later**. - The time curve is the **same shape, but phase shifted to the right** (appears delayed in time). **Sketch:** ``` | /‾‾‾‾\ /‾‾‾‾\ (vertical: p, horizontal: t) | / \ / \ |___/ \__/ \_____ <------T------> ``` *Alt text: The same sinusoidal over-pressure curve, shifted to the right along the time axis, with a horizontal arrow showing one period.* ### **Labels** - **Vertical axis:** Over-pressure, \( p(x, t) \) - **Horizontal axis:** Time, \( t \) - **Arrow:** One period, \( T \) --- # **Summary Table** | Graph | Horizontal Axis | Vertical Axis | Arrow Indicates | Notable Shift | |---------------|----------------|--------------|-------------------|-------------------------| | 1a (upper) | \( x \) | Over-pressure| \( \lambda \) | | | 1b (lower) | \( x \) | Over-pressure| \( \lambda \) | Shifted left | | 2a (upper) | \( t \) | Over-pressure| \( T \) | | | 2b (lower) | \( t \) | Over-pressure| \( T \) | Shifted right (delayed) | --- # **Final Answers** - **Pressure vs. position** at two times: Both are sinusoidal in \( x \), with the later graph shifted left by the distance the wave has traveled. One wavelength is marked. - **Pressure vs. time** at two positions: Both are sinusoidal in \( t \), with the graph at the farther position delayed in phase. One period is marked. **(See sketches and explanations above.)**

# 1. Pressure vs. Position at Fixed Time (Sound Wave Left) We analyze a **sound wave traveling to the left** along the \( x \)-axis, focusing on the over-pressure (pressure minus atmospheric pressure). ## Step-by-Step Solution ### a. Upper Graph: Pressure vs. Position at Time \( t_1 \) - **Axes:** - **Horizontal:** Position (\( x \)) - **Vertical:** Over-pressure (\( p(x, t_1) \)) - **Wave Equation:** For a wave traveling left, the equation is: \[ p(x, t_1) = A \cos(kx + \omega t_1 + \phi) \] This will be a cosine wave shape. - **Wavelength Arrow:** Indicate one full wavelength (\( \lambda \)) along the \( x \)-axis. ### b. Lower Graph: Pressure vs. Position at Later Time \( t_2 \) - At time \( t_2 > t_1 \), the wave has shifted left by a distance corresponding to the wave speed and time elapsed: \[ \text{Shift} = v \Delta t \] - The wave equation at \( t_2 \) becomes: \[ p(x, t_2) = A \cos(k(x + v \Delta t) + \omega t_2 + \phi) \] - **Wavelength Arrow:** Again, indicate one wavelength (\( \lambda \)). --- # 2. Pressure vs. Time at Fixed Position (Same Wave) Now we consider **pressure as a function of time** at a fixed point as the wave passes. ## Step-by-Step Solution ### a. Upper Graph: Pressure vs. Time at \( x_1 \) - **Axes:** - **Horizontal:** Time (\( t \)) - **Vertical:** Over-pressure (\( p(x_1, t) \)) - **Wave Equation:** At position \( x_1 \): \[ p(x_1, t) = A \cos(kx_1 + \omega t + \phi) \] This will also be a cosine wave shape. - **Period Arrow:** Indicate one period (\( T \)) on the time axis. ### b. Lower Graph: Pressure vs. Time at Position \( x_2 > x_1 \) - At position \( x_2 \), the wave arrives later than at \( x_1 \): \[ t_2 = t_1 + \frac{\lambda}{v} \] - The wave equation becomes: \[ p(x_2, t) = A \cos(kx_2 + \omega t + \phi) \] - **Period Arrow:** Again, indicate one period (\( T \)). --- # **Summary of Graphs** | Graph | Horizontal Axis | Vertical Axis | Arrow Indicates | Notable Shift | |---------------|----------------|--------------|-------------------|-------------------------| | 1a (upper) | \( x \) | Over-pressure| \( \lambda \) | | | 1b (lower) | \( x \) | Over-pressure| \( \lambda \) | Shifted left | | 2a (upper) | \( t \) | Over-pressure| \( T \) | | | 2b (lower) | \( t \) | Over-pressure| \( T \) | Shifted right (delayed) | --- # **Final Answers** - **Pressure vs. Position:** Two sinusoidal graphs with the later one shifted left. One wavelength is marked. - **Pressure vs. Time:** Two sinusoidal graphs with the later one delayed in time. One period is marked.