Let's say this is first mode,

this is the second mode,

and this is the third the mode.

So on and so on.

Then I can argue that, intuitively,

the vibration at any space along x-axis and any time,

can be superposition of these mode.

One plus 2, plus 3,

plus and so on.

For example plus n. There is

no other way that the vibration of a string,

vibration of a guitar string can

be the mode other than this.

But depending on the excitation,

the contribution of each mode will be different.

So I will say that is contribution of first mode is a_1,

second mode is a_2,

third mode is a_3,

nth mode is a_n plus.