Standing waves standing waves on a string: RESONANCE: reflection of wave at end of string, interference of outgoing with reflected wave “standing wave” nodes: string fixed at ends  displacement at end must be = 0  “(displacement) nodes” at ends of string  not all wavelengths possible; length must be an integer multiple of half-wavelengths: L = n /2, n = 1,2,3,… possible wavelengths are: n = 2L/n, n=1,2,3,… possible frequencies: fn = n  v/(2L), n=1,2,3,…. called “characteristic” or “natural” frequencies of the string; f1 = v/(2L) is the “fundamental frequency; the others are called “harmonics” or “overtones” RESONANCE: when a system is excited by a periodic disturbance whose frequency equals one of its characteristic frequencies, a standing wave develops in the system, with large amplitudes; at resonance, energy transfer to the system is maximal examples: pushing a swing; shape of throat and nasal cavity  overtones  sound of voice; musical instruments; Tacoma Narrows Bridge; oscillator circuits in radio and TV;