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The wave equation represents how waves propagate through space over time. For a wave, the standard form of the equation is:

• \( y = A \sin(kx \pm \omega t + \phi_0) \)

Here, \(A\) is the amplitude, \(k\) is the wave number, \(\omega\) is the angular frequency, and \(\phi_0\) is the initial phase. The wave equation describes properties like wavelength, frequency, and the direction in which the wave moves.
Within this context, the given equation \( y = A \sin(10 \pi x + 15 \pi t + \pi/3) \) helps us understand how different components of the wave interact. In this expression, \(10\pi\) and \(15\pi\) are closely related to the spatial and temporal characteristics, respectively. Understanding these components will allow us to determine other essential properties like wave direction and velocity.

The direction of a wave is analyzed based on the sign in front of the angular frequency term \(\omega t\). In the standard wave equation form, \( y = A \sin(kx \pm \omega t + \phi_0) \), the sign determines movement:

• A positive sign in front of \(\omega t\) (i.e., \(kx + \omega t\)) indicates a wave moving in the negative \(x\)-direction.
• A negative sign (i.e., \(kx - \omega t\)) signifies movement in the positive \(x\)-direction.

With the equation \( y = A \sin(10 \pi x + 15 \pi t + \pi/3) \), the wave moves in the negative \(x\)-direction. This results from the presence of a positive sign in front of the \(15\pi t\) term. LaTex formula or steps are unnecessary here but understanding the wave's equation gives powerful insights into its spatial behavior.

Wavelength is a crucial property of a wave which defines the distance over which the wave repeats itself. Wavelength \(\lambda\) is calculated from the wave number \(k\) with the formula:

• \( k = \frac{2\pi}{\lambda} \)

In our wave equation \( y = A \sin(10 \pi x + 15\pi t + \pi/3) \), the wave number \(k\) is \(10\pi\). Solving the relation:

• \( 10\pi = \frac{2\pi}{\lambda} \)
• \( \lambda = \frac{2\pi}{10\pi} \)
• \( \lambda = 0.2\, \text{m} \)

Thus, the wavelength of the wave is 0.2 meters. This concise calculation is essential to characterize the wave's spatial period and is important for wave-related applications.

The velocity of a wave indicates how fast the wave propagates through the medium. To find the wave velocity, one can use the relationship between angular frequency \(\omega\) and wave number \(k\):

• \( v = \frac{\omega}{k} \)

For our wave, where \(\omega = 15\pi\) and \(k = 10\pi\), substituting these values gives:

• \( v = \frac{15\pi}{10\pi} \)
• \( v = 1.5\, \text{m/s} \)

The wave velocity is 1.5 meters per second, indicating that the wave travels fairly quickly in the negative \(x\)-direction. Understanding wave velocity is fundamental for predicting how waves will interact with the environment or other waves.