Physics · Waves · NEET
Yes, they are the same speed written two ways. Start from v = omega / k. Put omega = 2 x pi x f and k = 2 x pi / lambda. Then v = (2 x pi x f) / (2 x pi / lambda) = f x lambda. NCERT writes this as v = omega/k = 2*pi*nu / (2*pi/lambda) = lambda/T = lambda*nu (nu is frequency). Use v = f x lambda when you are given frequency and wavelength; use v = omega / k when you are reading a wave equation y = a sin(kx - wt) and can pick out k and omega.
No, not for a given medium. Wave speed is fixed by the medium (for a string it is v = sqrt(T/mu); for sound in air it depends on temperature). So v stays the same. In v = f x lambda, if f goes up, lambda goes down by the same factor so that v does not change. Students often think higher frequency means faster wave, but that is the NEET trap. Speed is set by the medium, not by the source.
Read two numbers from the equation. The number multiplying x is k (angular wave number, unit rad/m). The number multiplying t is omega (angular frequency, unit rad/s). Then wave speed v = omega / k. Example: y = 2 sin(4x - 20t). Here k = 4 rad/m and omega = 20 rad/s, so v = 20 / 4 = 5 m/s. You do not need lambda or f separately.
In one full time period T, the wave pattern moves forward by exactly one wavelength lambda. Speed = distance / time = lambda / T. Since frequency f = 1 / T, this becomes v = lambda / T = lambda x f = f x lambda. This is why all three forms (f x lambda, lambda / T, omega / k) give the same number.
Wave speed v = f x lambda is how fast the wave shape travels through the medium, and it is constant for a given medium. Particle speed is how fast one particle moves up and down about its rest position; it keeps changing and its maximum value is a x omega (amplitude times angular frequency). They are different quantities. This is covered fully on the particle velocity vs wave velocity page.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
v = f x lambda, where f is frequency (in Hz) and lambda is wavelength (in metres). It gives speed in metres per second. From a wave equation it is written as v = omega / k.
Frequency f is in hertz (per second) and wavelength lambda is in metres, so v = f x lambda has units of metres per second (m/s). In v = omega / k, omega is in rad/s and k is in rad/m, which also gives m/s.
omega is angular frequency, omega = 2 x pi x f (unit rad/s). k is angular wave number or propagation constant, k = 2 x pi / lambda (unit rad/m). Their ratio omega / k is the wave speed.
For a fixed medium, yes. It depends on the medium's properties (tension and mass per length for a string; temperature for sound in air), not on the frequency of the source.
Use v = f x lambda when frequency and wavelength are given. Use v = omega / k when the question gives a wave equation like y = a sin(kx - wt), because you can read k and omega directly.
