To find the amplitude, wavelength, frequency, initial phase angle, displacement at time t = 0, and x = 0 for the wave specified by the equation y = 6 sin(2π(8t - 4x + θ)), let's break it down:
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. It can be determined by looking at the coefficient of 'x' in the equation. In this case, the coefficient of 'x' is -4. The wavelength is given by λ = 2π/|coefficient of x| = 2π/4 = π/2.
Frequency (f): The frequency represents the number of complete oscillations the wave undergoes in one unit of time. It is related to the wavelength by the equation f = 1/λ. Therefore, the frequency is f = 1/(π/2) = 2/π.
Initial phase angle (θ): The initial phase angle determines the starting point of the wave. In this equation, the initial phase angle is denoted by θ.
Displacement at time t = 0 and x = 0: To find the displacement at these points, we substitute t = 0 and x = 0 into the equation. For t = 0 and x = 0, the equation becomes y = 6 sin(2πθ). The value of y depends on the value of θ.
Please note that the equation you provided has an error with the parentheses around the argument of the sine function. It should be y = 6 sin(2π(8t - 4x + θ)).