In the context of the resultant equation of two traveling waves, the cos function with 2a inside it represents the amplitude of the resultant wave. The 2a term corresponds to the maximum displacement (amplitude) of the resulting wave.
When two waves with equal frequencies and similar amplitudes superpose, the resultant wave can be expressed as the sum of the two individual waves. The cos function in the equation describes the oscillatory behavior of the resultant wave, where the argument inside the cos function determines the frequency and phase relationship between the waves.
The sin function is often used in conjunction with the cos function to represent the phase difference between the waves. The phase difference determines the position of the resulting wave relative to the individual waves at any given point in time. The sin function helps incorporate this phase difference into the equation.
To better understand this concept, consider the general equation for two waves with equal frequencies and similar amplitudes, traveling in the same direction:
y = a * cos(kx - ωt + φ) + a * cos(kx - ωt)
In this equation, the first term represents one wave, and the second term represents the other wave. The cos functions with the arguments (kx - ωt + φ) and (kx - ωt) determine the positions and amplitudes of the waves at any given point in space and time.
The φ in the first term represents the phase difference between the waves. When φ is zero, the waves are in phase, meaning their crests and troughs align perfectly, resulting in constructive interference. If φ is non-zero, it represents a phase difference between the waves, resulting in a shift in the position of the resultant wave.
In summary, the cos function with 2a inside it represents the amplitude of the resultant wave, while the sin function, when used in conjunction with the cos function, represents the phase difference between the individual waves. These functions are used to describe the behavior and characteristics of the resultant wave when two traveling waves superpose.