https://file.aiquickdraw.com/mathbot/file/1729919811907eq57b10e.jpg

Given the wave equation provided in the image:

\[
y(x, t) = \frac{1}{2} P \sin(4 \pi x - P \pi t + \pi)
\]

and the instruction to set \( P = 10 \), we substitute \( P \) with 10, resulting in:

\[
y(x, t) = \frac{1}{2} \times 10 \sin(4 \pi x - 10 \pi t + \pi) = 5 \sin(4 \pi x - 10 \pi t + \pi)
\]

We will solve each part of the problem as follows:

### a) Fase Awal Gelombang (Initial Phase of the Wave)
The initial phase of the wave can be identified from the equation as the constant term in the argument of the sine function, which is \( \pi \).

### b) Fase pada Posisi \( x = 1.5 \) meter dan Waktu \( t = 0.25 \) detik
To find the phase at \( x = 1.5 \) meters and \( t = 0.25 \) seconds, substitute \( x = 1.5 \) and \( t = 0.25 \) into the argument of the sine function:

\[
\text{Phase} = 4 \pi (1.5) - 10 \pi (0.25) + \pi
\]
Calculate this expression to determine the phase at this specific point.

### c) Kecepatan Perambatan Gelombang (Wave Propagation Speed)
The wave propagation speed \( v \) for a sinusoidal wave of the form \( y(x, t) = A \sin(kx - \omega t + \phi) \) is given by:

\[
v = \frac{\omega}{k}
\]

In our equation, \( k = 4 \pi \) and \( \omega = 10 \pi \). Thus,

\[
v = \frac{10 \pi}{4 \pi} = \frac{10}{4} = 2.5 \, \text{m/s}
\]

### d) Kecepatan Transversal Maksimum (Maximum Transversal Speed)
The maximum transversal speed \( v_{\text{max}} \) is given by \( v_{\text{max}} = A \omega \), where \( A = 5 \) and \( \omega = 10 \pi \):

\[
v_{\text{max}} = 5 \times 10 \pi = 50 \pi \, \text{m/s}
\]

### e) Percepatan Transversal Maksimum (Maximum Transversal Acceleration)
The maximum transversal acceleration \( a_{\text{max}} \) is given by \( a_{\text{max}} = A \omega^2 \):

\[
a_{\text{max}} = 5 \times (10 \pi)^2 = 5 \times 100 \pi^2 = 500 \pi^2 \, \text{m/s}^2
\]

### f) Kecepatan Transversal di \( x = 2 \) meter dan \( t = 1 \) detik
To find the transversal speed at \( x = 2 \) meters and \( t = 1 \) second, we need to take the partial derivative of \( y(x, t) \) with respect to \( t \):

\[
\frac{\partial y}{\partial t} = 5 \cdot (-10 \pi) \cos(4 \pi x - 10 \pi t + \pi)
\]

Substitute \( x = 2 \) and \( t = 1 \) into this expression.

### g) Percepatan Transversal di \( x = 2 \) meter dan \( t = 1 \) detik
The transversal acceleration is found by taking the second derivative of \( y(x, t) \) with respect to \( t \):

\[
\frac{\partial^2 y}{\partial t^2} = -5 \cdot (10 \pi)^2 \sin(4 \pi x - 10 \pi t + \pi)
\]

Substitute \( x = 2 \) and \( t = 1 \) into this expression.

### h) Gambarkan Gelombang dengan Persamaan di Atas
To sketch the wave, plot \( y(x, t) = 5 \sin(4 \pi x - 10 \pi t + \pi) \) for different values of \( t \) or visualize the wave's profile over \( x \).

---

Would you like a detailed step-by-step calculation for each part, or help with specific parts? 

### Relative Questions to Explore Further
1. What is the relationship between frequency, wavelength, and wave speed?
2. How does the initial phase affect the wave at \( t = 0 \)?
3. How do transversal velocity and acceleration differ from wave speed?
4. What is the impact of changing \( P \) on the wave properties?
5. How can we interpret the wave’s graphical representation in terms of amplitude and wavelength?

### Tip
When calculating wave properties, always identify the wave number \( k \) and angular frequency \( \omega \) as they are key to determining speed and oscillatory motion.

Gelombang sinusoidal merambat pada tali dengan persamaan y(x, t) = (1/2) * P * sin(4πx - Pπt + π), dengan P diganti menjadi 10. Hitung fase awal gelombang, fase pada posisi x = 1.5 meter dan waktu t = 0.25 detik, kecepatan perambatan, kecepatan transversal maksimum, percepatan transversal maksimum, serta kecepatan dan percepatan transversal di x = 2 meter dan t = 1 detik.

Learn how to solve a sinusoidal wave equation with P set to 10. This includes calculating the initial phase, wave propagation speed, maximum transversal speed and acceleration, and specific transversal velocities and accelerations at given points. Suitable for undergraduate students studying physics or engineering.

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