# What is the theoretical/intutive meaning of $x(t) = x_0 \cos ( \omega t + f )$?

What is the meaning of $x(t)=x_0 \cos(\omega t+f)$, where $x_0$ is the amplitude, $\omega$ the angular frequency, $t$ time and $f$ the phase constant?

I know how to solve the mathematical problems which use this expression, but what I don't understand is how sine and cosine functions are used to express AC waves. I know the math but I do not understand the theory behind it. I do not have an intuitive understanding of these expressions. If you cannot explain at least guide me to a resource that can.

i want to understand the math in a theoretical manner like are we going to take the sine/cosine of the wt or the solved value of it, since sine/cos is a ratio so basically are we multiplying the since/cos fraction value to the variables, oh i am so confused please explain .

• I deleted some inappropriate comments. Please keep in mind that civility is expected at all times on Stack Exchange sites. Jun 20 '14 at 17:33

i want to understand the math in a theoretical manner like are we going to take the sine/cosine of the wt or the solved value of it, since sine/cos is a ratio so basically are we multiplying the since/cos fraction value to the variables, oh i am so confused please explain .

The quantity $\omega$ represents an angular velocity, so $\omega t$ is an angle, and taking trigonometric functions of angles is a perfectly natural thing to do.

The next question is "What do these angles represent?". Enter the "phasor" (which, alas, has nothing to do with Star Trek). A phasor is a vector which is abstractly associated with some physical quantity and has the following properties

• It's magnitude is the maximum value of the physical quantity it is associated with.
• It rotates at a steady angular velocity denoted $\omega$ which you have to select to be appropriate to the situation.
• At any given time it's projection onto one of the coordinate axis (and which one depends on the book you read, sorry) is the current value of the physical quantity represented. Notice that depending on the axis you choose the project takes the form of either a sine or a cosine and that the resulting time-dependence has the form you exhibit in the question.

As an example of how to select omega, consider the electricity that comes out of your wall: the frequency is 60 Hz (US, Japan and others) or 50 Hz (British commonwealth and others) making the angular velocity $\frac{60}{2\pi} \,\mathrm{rad/s}$ or $\frac{50}{2\pi} \,\mathrm{rad/s}$.

• can you tell me of a resource wherein i can learn all what you have said from Jun 19 '14 at 16:24
• The chapter on (non-trivial) AC circuits in most first year physics books should have this. Even many of the algebra/trig based books like Cutnell&Johnson's Physics go over it. It will often be associated with the analysis of LRC (or LCR) circuits (those containing resistance, inductance and capacitance). Jun 19 '14 at 19:55

Waves such as alternating current are expressed as a combination of cosine and sine functions because they are periodic. All periodic functions can be expressed as fourier series, i.e. as a sum of sine and cosine. The more complex the periodic oscillation, the more terms are required. Since AC is a simple oscillation, only a small number of terms are needed. See: http://en.wikipedia.org/wiki/Fourier_series

• i know all that i mean to understand this in an intutive manner Jun 19 '14 at 15:45
• Do you mean to ask why is it a periodic function or why periodic functions can be represented as fourier series? Jun 19 '14 at 15:47
• yes something like that, i know the functions follow the sine wave but not really are sine, but still you understood what i meant. i am asking what is the meaning of sine of a variable. oh i think i need to explain myself better. Jun 19 '14 at 15:49

You mentioned phase constant, angular frequency and time. So, it must be a wave function. But I did not understand why you mentioned cos function since wave is a sine function.

To understand waves, we should understand simple harmonic motion(SHM) first. It is a type of to and fro periodic motion. Velocity and thereby kinetic energy of object undergoing SHM is nil at two extreme positions and maximum at mean position. Potential energy is maximum at extreme positions and nil at mean position. So, it can be thought of as conversion of two forms of energy in pace of time. Motion of pendulum is a example of SHM. The motion is so uniform (continuous in mathematics) that it follows precise mathematical model i.e. sine function.

Projection of uniform circular motion along it's diameter was studied by mathematicians and it was obviously a sin function. Amazingly, it exactly matched with motion of simple pendulum. You can set simple pendulum into motion, note it's period then you can calculate displacement from mean position at any point of time fitting values in 'motion of diameter projection of circle' formula.

Due to similarity of diameter projection, to and fro motion of simple pendulum is sin function.

Now, come to waves part. Waves transfer energy by oscillation of medium. Medium oscillates exactly like simple pendulum. Wave function is composed of two parts. Amplitude function and phase function multiplied together. First is amplitude part. It is constant, so no need to consider as function of any variable. Maximum displacement(extreme point)from mean position is called amplitude (A) which is multiplied ahead of sine function. Second is phase part. Phase ranges from -1 to +1. You can make analogy of phase part with unit vector which provides direction with magnitude of 1. Omegatime means angular velocitytime. Angular velocity remains constant. So, the only variable is time.

One important point to understand is you can not directly compare SHM of particle with angular velocity and time.(concept of angular velocity applies only in circular motion.)So,first reverse project SHM of wave particle into circular motion as we did in first part of answer, then you will be able to relate SHM with angular velocity *time.

I explain phase difference now. Suppose two particles represent two separate parallel sound waves. One is in mean position and other is in extreme positions, there will be 180 degree phase difference if we project it into circle. Their energy (Kinetic energy of first and potential energy of second) get subtracted. Conversely if both are at mean position or both at extreme position, energy gets added. You can imagine in between the extremes.

Note that omega*time is angle and fi is also angle. So two angles can be added.