The displacement u ( x , t ) for the equation u ( x , t ) = ∑ n = 1 ∞ c n sin ( n π x L ) cos ( n π a t L ) for the solution of the string problem with zero velocity is E ( t ) = π 2 T 4 L ∑ n = 1 ∞ n 2 c n 2 .

Question

Want to see the full answer?

Answer 1

Question

Chapter 10.7, Problem 22P

To determine

To show: The displacement u(x,t) for the equation u(x,t)=∑n=1∞cnsin(nπxL)cos(nπatL) for the solution of the string problem with zero velocity is E(t)=π2T4L∑n=1∞n2cn2.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Students have asked these similar questions

Temperature measurements are based on the transfer of heat between the sensor of a measuring device (such as an ordinary thermometer or the gasket of a thermocouple) and the medium whose temperature is to be measured. Once the sensor or thermometer is brought into contact with the medium, the sensor quickly receives (or loses, if warmer) heat and reaches thermal equilibrium with the medium. At that point the medium and the sensor are at the same temperature. The time required for thermal equilibrium to be established can vary from a fraction of a second to several minutes. Due to its small size and high conductivity it can be assumed that the sensor is at a uniform temperature at all times, and Newton's cooling law is applicable. Thermocouples are commonly used to measure the temperature of gas streams. The characteristics of the thermocouple junction and the gas stream are such that λ = hA/mc 0.02s-1. Initially, the thermocouple junction is at a temperature Ti and the gas stream at…

A body of mass m at the top of a 100 m high tower is thrown vertically upward with an initial velocity of 10 m/s. Assume that the air resistance FD acting on the body is proportional to the velocity V, so that FD=kV. Taking g = 9.75 m/s2 and k/m = 5 s, determine: a) what height the body will reach at the top of the tower, b) how long it will take the body to touch the ground, and c) the velocity of the body when it touches the ground.

A chemical reaction involving the interaction of two substances A and B to form a new compound X is called a second order reaction. In such cases it is observed that the rate of reaction (or the rate at which the new compound is formed) is proportional to the product of the remaining amounts of the two original substances. If a molecule of A and a molecule of B combine to form a molecule of X (i.e., the reaction equation is A + B ⮕ X), then the differential equation describing this specific reaction can be expressed as: dx/dt = k(a-x)(b-x) where k is a positive constant, a and b are the initial concentrations of the reactants A and B, respectively, and x(t) is the concentration of the new compound at any time t. Assuming that no amount of compound X is present at the start, obtain a relationship for x(t). What happens when t ⮕∞?