If f and g are twice differentiable real functions of a single variable, show that the functionu(x, t) = f (x + at) + g(x − at)is a solution of the ‘wave equation’ utt = a2uxx

Aerospace Engineering (AE)Page 9 of 35Organizing Institute: IIT KanpurQ.13 Consider the one-dimensional wave equation0u ut x + =  forx−    ,0t  .For an initial condition2( ,0) xu x e−= , the solution at1t = is(A)2( 1)( ,1) xu x e− −=(B)1( ,1)u x e−=(C)2( ,1) xu x e−=(D)2( 1)( ,1) xu x e− +=Q.14 A two-dimensional potential flow solution for flow past an airfoil has a streamlinepattern as shown in the figure. Which of the following conditions is additionallyrequired to satisfy the Kutta condition?(A) Addition of a source of strength0Q (B) Addition of a source of strength0Q (C) Addition of a circulation of strength0  (counter-clockwise)(D) Addition of a circulation of strength0  (clockwise

The general solution of the wave equation by the sum of eigenfunctions is a result of:Group of answer choicesThe fundamental theorem of calculusThe fundamental theorem of superpositionThe convolution theorem

If f and g are two functions of x, then

Schrodinger wave equation

The function u(x,t)=ei(x+ct)𝑢(𝑥,𝑡)=𝑒𝑖(𝑥+𝑐𝑡) is another solution to the wave equation∂2u∂t2−c2∂2u∂x2=0∂2𝑢∂𝑡2−𝑐2∂2𝑢∂𝑥2=0  Let's investiage the case when c=2+5i𝑐=2+5𝑖 .  ThenThe real part of u𝑢 is    The imaginary part of u𝑢 is   .So the function oscillates in the complex plane, but these oscillations are dampened by the exponential term. As time goes onlimt→∞u(x,t)=lim𝑡→∞𝑢(𝑥,𝑡)=     .