ESCI 343 – Atmospheric Dynamics II Answers to Selected Exercises for Lesson 4 – Introduction to Waves 3. A wave is represented in complex notation as ()tkxiAetxuω−=),( where iA 32−=. Show that this is equivalent to representing the wave as ()()tkxtkxtxuωω−+−= sin3cos2),(. Answer: Let tkxωθ−=. Then, ()()( )θθθθθθθθθθcos3sin2sin3cos2sin3cos3sin2cos2sincos322−++=−+−+=+−=iiiiiiu and we’re only interested in the real part, so θθsin3cos2+=u 4. Find the phase difference between the following two waves, ()( )tkxitkxiBetxvAetxuωω−−==),(),( for the following values of A and B. a. iBiA 23;32+−=+= Answer: A ∝ iB; 270° b. iBiA 32;32−−=+= Answer: A ∝ −B; 180° c. iBiA 23;32−=+= Answer: A ∝ −iB; 90° d. iBiA 64;32+=+= Answer: A ∝ B; 0° e. iBiA 69;32−=+= Answer: A ∝ −iB; 90°25. a. Let a wave be represented by ikxexu =)( . Show that u and du/dx are 270° out of phase. Answer: iuikudxdu∝= b. Let a wave be represented by kxxucos)(=. Show that u and du/dx are 270° out of phase, which shows the consistency of representing sinusoids using complex notation. Answer: kxkdxdusin−= , and cos x and −sin x are 270° out of phase. 9. What is the physical meaning of a complex frequency? In other words, if ω has an imaginary part, what does this imply? Hint: Put iriωωω+= into ()tkxieuω−= and see what you get. Answer:()ttkxiireeuωω−= , so waves amplify or decay exponentially with time. 10. Show that Κ∂∂=ωgc . Answer: