Given below are the some of the function of x and t to represent the displacement
(transverse or longitudnal ) of an elastic wave . State whether it is stationary or travelling
wave or none of them.
(i) y = 3sin(5x+0.5t) + 4cos(5x+0.5t)
(ii) y = 3sin(5x+0.5t) + 4cos(5x-0.5t)
(iii) y = cosxsint + cos2x/sin2t
(iii) y = 3sin(5x-0.5t) + cos2xsin2t
To determine whether each given function represents a stationary wave, a traveling wave, or none of them, we need to analyze the form of the functions and how they depend on both x (position) and t (time).
(i) y = 3sin(5x+0.5t) + 4cos(5x+0.5t)
In this function, both sin(5x+0.5t) and cos(5x+0.5t) depend on the sum of 5x and 0.5t. Since both terms are added together and have the same arguments, it indicates that the wave is a traveling wave. Therefore, (i) represents a traveling wave.
(ii) y = 3sin(5x+0.5t) + 4cos(5x-0.5t)
Similar to the previous function, here sin(5x+0.5t) and cos(5x-0.5t) depend on the sum and difference of 5x and 0.5t, respectively. This indicates that the wave is a traveling wave as well. Therefore, (ii) represents a traveling wave.
(iii) y = cosx*sin(t) + cos(2x)/sin(2t)
In this function, the terms cosx*sin(t) and cos(2x)/sin(2t) have fixed values and do not depend on either x or t. The wave is not changing with either position or time. Hence, this type of wave is called a stationary wave. Therefore, (iii) represents a stationary wave.
(iv) y = 3sin(5x-0.5t) + cos(2x)sin(2t)
Similar to the previous cases, sin(5x-0.5t) and cos(2x)sin(2t) depend on the difference and product of 5x and 0.5t, respectively. This indicates that the wave is a traveling wave. Therefore, (iv) represents a traveling wave.
To summarize:
(i) Traveling wave
(ii) Traveling wave
(iii) Stationary wave
(iv) Traveling wave
To determine whether each given function represents a stationary wave, a traveling wave, or none of them, we need to analyze the dependence of the displacement on variables x and t.
(i) y = 3sin(5x+0.5t) + 4cos(5x+0.5t)
This function represents a traveling wave because both sine and cosine terms have the same arguments (5x+0.5t), indicating that the wave is propagating in the positive x-direction.
(ii) y = 3sin(5x+0.5t) + 4cos(5x-0.5t)
This function also represents a traveling wave. The different signs in the arguments of sine and cosine terms suggest that the wave is traveling in the negative x-direction.
(iii) y = cos(x)sin(t) + cos(2x)/sin(2t)
This function does not represent either a stationary or traveling wave because it contains both sine and cosine terms with independent arguments x and t. The terms do not represent any specific wave motion.
(iv) y = 3sin(5x-0.5t) + cos(2x)sin(2t)
This function represents a traveling wave because both the sine and cosine terms have the same arguments (5x-0.5t, 2x, 2t), indicating that the wave is propagating in the positive x-direction.
To summarize:
(i) Traveling Wave
(ii) Traveling Wave
(iii) None of them
(iv) Traveling Wave