What are the low points of a transverse wave called, and what do they signify?

The low points of a transverse wave are called troughs. In a transverse wave, the particles of the medium move perpendicular to the direction of the wave's propagation. The troughs are the points where the particles are at their maximum displacement in the direction opposite to that of the crest, which is the high point of the wave.

The significance of troughs in a transverse wave can be understood in terms of the wave's amplitude and energy. The amplitude of a wave is the maximum displacement of the particles from their equilibrium position, and it is measured from the equilibrium position to a crest or from the equilibrium position to a trough. The amplitude is related to the energy carried by the wave; a larger amplitude means the wave is carrying more energy.

To visualize a transverse wave, imagine a rope held at one end and flicked up and down at the other. The wave that travels through the rope has high points (crests) and low points (troughs). The distance between two consecutive crests or two consecutive troughs is known as the wavelength (\(\lambda\)), which is one of the fundamental properties of a wave.

The mathematical representation of a transverse wave traveling along the x-axis can be given by the wave function:

\[ y(x,t) = A \sin(kx - \omega t + \phi) \]

where:
- \(y(x,t)\) is the displacement of the wave at position \(x\) and time \(t\),
- \(A\) is the amplitude of the wave,
- \(k\) is the wave number, related to the wavelength by \(k = \frac{2\pi}{\lambda}\),
- \(\omega\) is the angular frequency of the wave, and
- \(\phi\) is the phase constant.

At the troughs, the sine function has a value of -1, indicating that the displacement \(y(x,t)\) is at its minimum value, which is -\(A\). This represents the point of maximum negative displacement from the equilibrium position.

In summary, troughs are the lowest points in a transverse wave, indicating the maximum negative displacement of the medium's particles. They are crucial for understanding the wave's amplitude and the energy it transports.