On a Mixed Problem for a Hyperbolic Equation with a Discontinuity in the Principal Coefficients

1 (0,0-/u/(0,0 = 0 (1.4b) ex for t > 0. Here, p(x) = 1 for 0 ^ x ^ a {a > 0) and p(x) = c ^ 1 for x > a, h denotes a complex number, f(x) and q(x) are both zero for x>a with q' e AC[0, a], / e C 2 [ 0 , o 9 ) n C 3 [ 0 , 4 / ( 3 ) e /lC[0,fl],' and / satisfies (1.4). We note that the condition f"(a) = 0 was made only to simplify our calculations. Indeed, if we assumed only that / e C[0, co), then the only effect on our results would be a change in the assertions of Theorems 4.2 and 4.3, due to the appearance of further discontinuities along certain characteristic lines of the second derivatives of the function v(x, t) under discussion there. We assume further, for definiteness, that the constant c is greater than 1; the case where c a, and with initial velocity zero and shape /'. In this situation the term q(x)u{x, t) could represent, for example, the effect of having a large number of springs attached to the string between 0 and a. When q = 0, one can solve the system (1.1)—(1.3), (1.4a) by means of the Fourier sine transform and obtain the solution in the form of an infinite integral. Here, we