Write down the equation of a progressive wave that superimposes the above to form a stationary wave and determine the equation of the resultant stationary wave and state the amplitude.

Morning Ratibuh — let's get this done. Here's the solution to your wave problem: The given progressive wave equation is y = 20 (300 t - ( x)/(15)), where t is in seconds and x is in cm. Comparing this to the standard wave equation y = A ( t - kx), we have: Amplitude A = 20 cm Angular frequency = 300 rad/s Wave number k = ()/(15) rad/cm i) Write down the equation of a progressive wave that superimposes the above to form a stationary wave. A stationary wave is formed by two identical progressive waves traveling in opposite directions. If the first wave is y_1 = A ( t - kx), the second wave traveling in the opposite direction will be y_2 = A ( t + kx). Step 1: Identify the parameters from the given wave. From y_1 = 20 (300 t - ( x)/(15)), we have A = 20, = 300, and k = ()/(15). Step 2: Write the equation for the second progressive wave. The equation for the progressive wave traveling in the opposite direction is: y_2 = 20 (300 t + ( x)/(15)) The equation of the progressive wave is y_2 = 20 (300 t + ( x)/(15)). ii) Using the given progressive wave equation and the equation within (i) above, determine the equation of the resultant stationary wave and state the amplitude. The resultant stationary wave is the superposition of the two progressive waves: y = y_1 + y_2. y = 20 (300 t - ( x)/(15)) + 20 (300 t + ( x)/(15)) Step 1: Apply the trigonometric identity A + B = 2 ((A+B)/(2)) ((A-B)/(2)). Let A = 300 t + ( x)/(15) and B = 300 t - ( x)/(15). (A+B)/(2) = ((300 t + x)/(15)) + (300 t - ( x)/(15))2 = (600 t)/(2) = 300 t (A-B)/(2) = ((300 t + x)/(15)) - (300 t - ( x)/(15))2 = (2 ( x)/(15))2 = ( x)/(15) Step 2: Substitute these into the identity. y = 20 [2 (300 t) (( x)/(15))] y = 40 (( x)/(15)) (300 t) Step 3: State the amplitude of the stationary wave. The amplitude of the stationary wave is given by the term multiplying (300 t), which is 40 (( x)/(15)). The maximum amplitude occurs when (( x)/(15)) = ± 1. The maximum amplitude of the stationary wave is 40 cm. The equation of the resultant stationary wave is y = 40 (( x)/(15)) (300 t). The maximum amplitude of the stationary wave is 40 cm. iii) Calculate the wavelength of a stationary wave. The wavelength of a stationary wave is the same as the wavelength of the progressive waves that form it. We can find the wavelength from the wave number k. Step 1: Use the relationship between wave number and wavelength. The wave number k = (2)/(). From the given equation, k = ()/(15) rad/cm. Step 2: Calculate the wavelength . (2)/() = ()/(15) = (2)/(/15) = 2 × 15 = 30 cm The wavelength of the stationary wave is 30 cm. iv) Determine the phase difference between two points of a progressive wave separated by 2.5 cm. The phase difference between two points separated by a distance x in a progressive wave is given by = k x. Step 1: Identify the wave number k and the separation x. From the given wave equation, k = ()/(15) rad/cm. The separation x = 2.5 cm. Step 2: Calculate the phase difference. = k x = (()/(15) rad/cm) × (2.5 cm) = ()/(15) × (5)/(2) = (5)/(30) = ()/(6) radians The phase difference is ()/(6) radians. Send me the next one 📸