Choosing a simple parallel-plate waveguide, an exact expression is obt a i n e d for t h e t r a n s i e n t response w h e n a s t e p -m o d u l a t e d carrier signal is applied. T h e analysis c o n s t i t u t e s a m o d e s t extension of t h e early work of Sommerfeld a n d t h e more recent investigations of Rubinowicz a n d Knop. N u m e r i c a l calculations are p r e s e n t e d for a range of t h e p a r a m e t e r s which h a v e some relevance to p r o p a g a t i o n in t h e e a r t h -i o n o s p h e r e waveguide. The " e x a c t " form of t h e t r a n s i e n t envelope function is c o m p a r e d w i t h an app r o x i m a t e version which is in t h e form of a Fresnel integral. I t is shown t h a t t h e a p p r o x i m a t e m e t h o d gives a good q u a l i t a t i v e e s t i m a t e for t h e t r a n s i e n t response characteristics. Thus, confidence is gained in applying it to o t h e r situations where an exact solution is n o t available. ยง 1. Introduction. In many instances, one is called upon to estimate the influence of the propagation medium on the received waveform of a transient signal. This is the topic of the present paper. To be definite, we designate the waveform of a component of the electromagnetic field at the input to be co(t). Then, the problem is to calculate the waveform of the corresponding component e(t) at the output by making use of the specified spectral properties of the propagation channel.