In the wave equation y = 6 sin(2π(8t - 4x + θ)), we can determine the following parameters:
Amplitude (A): The amplitude represents the maximum displacement of the wave from its equilibrium position. In this case, the amplitude is given as 6.
Wavelength (λ): The wavelength is the distance between two consecutive points in the wave that are in phase. To find the wavelength, we need to compare the equation with the general form of a sinusoidal wave, which is y = A sin(2π(x/λ - t/T + φ)), where T is the period and φ is the phase angle. Comparing the equations, we have:
2π(8t - 4x + θ) = 2π(x/λ - t/T + φ)
Comparing the corresponding terms, we find: x/λ = 1/4 (from 4x) t/T = 1/8 (from 8t) φ = -θ/2π (from θ)
From x/λ = 1/4, we can conclude that λ = 4 units.
Frequency (f): The frequency is the number of complete cycles (oscillations) of the wave per unit of time. It is the reciprocal of the period. In this case, the coefficient of 't' in the wave equation is 8, so the frequency is 8 cycles per unit of time.
Initial phase angle (θ): The initial phase angle determines the position of the wave at time t = 0 and x = 0. In this case, the phase angle is given as θ.
Displacement at time t = 0 and x = 0 (y₀): To find the displacement at t = 0 and x = 0, we substitute these values into the wave equation:
y₀ = 6 sin(2π(8(0) - 4(0) + θ)) = 6 sin(2πθ)
Therefore, the parameters of the given wave are: Amplitude (A) = 6 Wavelength (λ) = 4 Frequency (f) = 8 Initial phase angle (θ) = θ Displacement at time t = 0 and x = 0 (y₀) = 6 sin(2πθ)