Final answer:
The velocity of a piece of the string in the given wave equation is found by differentiating the displacement with respect to time. The resulting expression is A(2πv/λ) sin((2π/λ) (x - vt)), which represents the instantaneous velocity at position x and time t.
Step-by-step explanation:
To find the expression for the velocity at which a piece of the string moves in the given wave, we need to differentiate the displacement u(x, t) = A cos((2π/λ) (x - vt)) with respect to time t. The partial derivative of u with respect to t is
∂u/∂t = A(∂/∂t)cos((2π/λ) (x - vt))
To differentiate the cosine function with respect to t, we use the chain rule. The derivative of the cosine function is -sin, and the inside function (2π/λ)(-v) is differentiated to be -2πv/λ.
∂u/∂t = - A sin((2π/λ) (x - vt)) (∂/∂t)(2π/λ)(x - vt) = - A sin((2π/λ) (x - vt))(-2πv/λ) = A(2πv/λ) sin((2π/λ) (x - vt))
This expression represents the instantaneous velocity of a piece of the string at position x and time t.
The displacement of a tight string, u(x, t), is given by the equation u(x, t) = A cos((2π/λ)(x - vt)). To find the expression for the velocity (∂u/∂t) at which a piece of the string moves, we need to differentiate the displacement equation with respect to time, t. Differentiating u(x, t) = A cos((2π/λ)(x - vt)) with respect to t gives us:
∂u/∂t = -A(2π/λ)v sin((2π/λ)(x - vt))
Therefore, the expression for the velocity (∂u/∂t) at which a piece of the string moves is -A(2π/λ)v sin((2π/λ)(x - vt)).