I'm currently reading about forced oscillations, and in the book (A course in Classical Physics by Alessandro Bettini) I'm using, they start with the equation

$$\frac{d^2x}{dt^2} + \gamma\frac{dx}{dt} + \omega_0^2 x = \frac{F_0}{m} \cos{\omega t} \, .$$

They then solve for the stationary solution $$x_s(t)=B\cos{(\omega t-\delta)} \, ,$$


$$\delta = \arctan \left( \frac{\gamma \omega}{\omega _0^2 - \omega^2} \right)$$

is said to be the "phase delay of the displacement $x$ relative to the instantaneous phase of the force". However, I'm afraid I'm having trouble understanding what this statement means. What exactly do they mean by "phase delay" and how does this term actually relate to the displacement and the driving force?

  • $\begingroup$ the driving is Acos(wt) and the response is in Bcos(wt+delta). if you plot those two function, you'll see there is a phase difference bewteen those two cosine signal. This is your phase 'delay' $\endgroup$
    – sailx
    Jun 15, 2017 at 12:15
  • $\begingroup$ Oh ok, that makes a lot more sense to me now. Thank you so much. $\endgroup$ Jun 15, 2017 at 12:29
  • $\begingroup$ By the way, if the oscillator is excited at its resonance frequency $\omega_0$, then the argument of the arctan get's infinite, as a consequence $\delta =\pm 90^{\circ}$. Therefore in case of a resonant excitation the phase shift is $90^{\circ}$ (Actually, the $\omega$ which is used in your differential equation on the left side should be different from the $\omega$ used on the right side in general). $\endgroup$ Jun 15, 2017 at 13:35
  • $\begingroup$ @FredericThomas You're right, I made a typo and have fixed it. $\endgroup$ Jun 15, 2017 at 14:29
  • $\begingroup$ The author of this answer believes it is especially. relevant. $\endgroup$
    – rob
    Oct 23, 2018 at 18:53

2 Answers 2


Imagine you are holding to top is the spring in a spring-mass system and that the mass is in water to produce a reasonable amount of damping.
Now imagine moving your hand up and down very, very slowly.
The mass will almost exactly follow the motion of your hand - essentially the mass and your hand are in phase with one another.
Now imagine your hand moving up and down with the same amplitude as before at a greater frequency.
The amplitude of oscillation of the mass will be larger and it will lag behind the motion of your hand.
In simple terms you can think of it as the mass taking time to react to what your hand is telling it to do.
At an even higher frequency of oscillation of your hand the amplitude of oscillation of the mass will increase but it will lag further behind the motion of your hand until at a high enough frequency of oscillation of your hand the amplitude of motion of the mass will be a maximum and the motion of the mass will lag by $90^\circ$ - you have resonance.
Increasing the frequency of motion of your hand results in the amplitude of motion of the mass being reduced and the phase lag between the hand and the mass increased.
At very high frequencies of your hand the mass hardly moves at all and it's motion is approximately $180^\circ$ behind the motion of your hand.

You can do the demonstration described above but a much clearer demonstration is shown here using a variable speed motor driving a wheel attached off centre via a string and pulley arrangement to the top of a spring.
The transients are damped down with the aid a circular piece of card attached to the mass which produces a reasonable amount of friction due to the air.
The difference in phase (and the amplitude) between the driver (motor and wheel) and the driven mass is very clearly seen at different speeds of the motor.


This is an example of forced oscillation.Normally, we expect and have observed that if an external force is oscillating with a particular frequency, then the object will also start oscillating with the same frequency after some time.This is called the steady state solution.

Of course,it doesn't start that way.At starting,we see that there are some transient solutions which can be obtained by solving this differential equation to obtain a general solution . Steady state solution isn't implemented immediately because the system has an inertia and wants to stay in it's original state. But if the force is provided for a very long time, these transients die down but leave behind a phase. Suppose you want to swing a pendulum with a particular frequency and you apply a periodic force by your hand.The system will also start oscillating with your frequency but not in the same phase as your hand is moving.


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