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Transverse waves are a type of wave where the oscillations occur perpendicular to the direction of the wave's travel. Imagine a rope being flicked up and down; the motion of the rope is up and down, while the wave travels horizontally along the length of the rope.

Transverse waves can be seen in many scenarios including light waves and water ripples. In distinguishing them from longitudinal waves, like sound waves, transverse waves require a medium that supports shear stress, such as solids. This is part of the reason why transverse waves do not travel through gases or liquids effectively — they don't support the sideways shear that these waves create.

In our wave equation, the wave is transverse since it involves the displacement of the medium (the rope) perpendicular to the wave's direction.

The wavelength of a wave is a fundamental property and gives the distance over which the wave's shape repeats. It is represented by the symbol \( \lambda \) and is typically measured in meters or centimeters. Wavelength is inversely proportional to the wave number (k), according to the formula: \[\lambda = \frac{2\pi}{k} \]

In this specific wave equation, we have identified the wave number as \(0.400\pi \ \, \text{cm}^{-1} \). Therefore, substituting into the wavelength formula gives us: \[\lambda = \frac{2\pi}{0.400\pi} \approx 5 \, \text{cm} \]

This implies that every 5 cm along the rope, the wave pattern repeats itself, indicating the regular periodic property of the wave.

Wave speed \( v \) is a crucial concept in understanding how fast a wave propagates through a medium. It is calculated by multiplying the frequency \( f \) by the wavelength \( \lambda \): \[v = f \times \lambda\]

In our example, the frequency was found to be 125 Hz, and the wavelength is 5 cm. This results in a wave speed of:
\[ v = 125 \, \text{Hz} \times 5 \, \text{cm} = 625 \, \text{cm/s} \]

Thus, the wave travels at a speed of 6.25 m/s along the rope. Wave speed aligns with how quickly energy and information are transmitted through the medium.

Wave power pertains to the rate at which energy is transferred by the wave. For mechanical waves like those on a rope, the average power \( P_{avg} \) can be calculated using: \[P_{avg} = \frac{1}{2} \mu A^2 \omega^2 v \]

Where:

• \( \mu \) is the mass per unit length of the rope.
• \( A \) is the amplitude of the wave in meters.
• \( \omega \) is the angular frequency.
• \( v \) is the wave speed.

In our specific example, substituting the values gives us: \

• \( \mu = 0.0500 \, \text{kg/m}, \)
• \( A = 0.0075 \, \text{m}, \)
• \( \omega = 250\pi \, \text{rad/s}, \)
• \( v = 6.25 \, \text{m/s} \)

\[ P_{avg} \approx 17.3 \, \text{W} \]

This calculation represents the energy transmitted by the wave per second, giving us insight into how much work can be done by the energy carried in the wave.