2. REFLECTION OF WAVES FROM DISCONTINUITIES ON A STRING 



Consider a uniform string of line density p ly*'ig along the 



portion -— <x < of rhe x-axls under tension T. Consider this string 



to be joined at x ■ by a massless connection to a string of density Q l 



lying along < x < +■>, the tension there also being T. Transverse 



> 

 waves of arbitrary profile propagate at speed C ■ (T/p ) on the 



JL ° ° 

 left-hand it ring, and at speed C x - (T/p ) 2 on the right-hand. 



Alternatively, with time dependence exp(-iut), u> > 0, understood through- 

 out, waves in x < have wavenumber k ■ u/C while those in x > 



o o 



nave wavenunber k » u/C . We wish to determine the reflected and trans- 

 Bitted waves In x < 0, x > respectively when a progressive wave with 

 displacement exp(ik x) is incident upon the junction from x < 0. [See 

 Figure 1.] 



This is a trivial problem to which the solution can be found by 

 elementary aethods and which can be generalized to cover a variety of 

 different conditions at the junction. It and its generalizations are 

 also suitable for Introducing the W-H method very simply. We start 

 by writing the total displacement as 



y + exp(ik x) in x < | 



and as «'» 



y in x > / 



The reason for this is that then y (which might be called the scattered 

 field) must take the form of an outgo ing wave ass* +°°aad as x *■», 

 and this enables us to state something about the Fourier transform of y. 

 We have first, however, to make t!j wavenumbers k and k slightly 



