Intuition The one core ideaA wave equation says "a bump on a string moves without changing shape," and the whole topic is built from one trick: a partial derivative measures how fast something changes when you wiggle only ONE input. Master that single picture — a slope you read while holding all-but-one variable frozen — and every symbol on the parent page (the ∂ ξ , η
This page assumes nothing . If the parent note wrote a symbol, we build it here from a picture first. Read top to bottom; each block earns the next.
f ( x ) machine that eats one number x f ( x ) curve : horizontal axis = the input x
On the parent page f ( x ) = e − x 2
The string, frozen at time zero, IS such a curve: height above each point x f
We keep saying "slope." Let us earn it.
Definition Slope / derivativeStand at a point on the curve and lay a tiny straight ruler flat against it — the tangent line . How steeply that ruler tilts is the slope . Written f ′ ( x ) d x df f
Intuition Why a slope answers a real questionThe slope answers "if I nudge x Steep uphill = big positive slope; flat = zero; downhill = negative. That "rate of change" is the only idea derivatives ever encode.
We only ever use f ′
u ( x , t ) two numbers: position x t surface — a landscape whose height above the floor point ( x , t ) u
Intuition What it means physicallyx t u
Every symbol from here on lives on this landscape.
The whole subject is called Partial Differential Equations for one reason:
Definition Partial derivativeOn a two-input landscape you cannot ask "the slope" — slope in which direction? So we freeze one input and slide only the other, then read the ordinary slope. Freezing t x partial derivative u x ∂ x u ∂ x ∂ u x t u t
Slice the landscape with a vertical knife. Cut along the x (time held still): you expose a curve, and its slope is u x along the t (place held still): its slope is u t velocity .
That is exactly why the parent's second initial condition u t ( x , 0 ) = g ( x ) initial velocity : u t time .
∂ x d x d Why it feels right: both measure a slope in x Reality: the curly ∂ second variable (t d
u xx u x its partial in x curvature , how sharply the string bends. Likewise u tt acceleration of a point.
Intuition Reading the wave equation as a sentenceu tt = c 2 u xx "a point accelerates (u tt u xx A sharply curved spot snaps back hard — that is what makes waves ripple. The number c 2
Definition The wave speedc c > 0 speed a bump travels, in length-per-time. On the picture it is the tilt of the characteristic lines you meet next.
Definition Differential operator∂ x verb : "differentiate whatever comes next in x ∂ t + c ∂ x ∂ t c ∂ x ∂ tt − c 2 ∂ xx = ( ∂ t − c ∂ x ) ( ∂ t + c ∂ x ) .
Intuition Why factoring isallowed
Because these verbs behave like ordinary algebra: a 2 − b 2 = ( a − b ) ( a + b ) a , b commute (order doesn't matter). For smooth u u x t = u t x ∂ x ∂ t transport equations .
ξ = x − c t , η = x + c t new coordinates built from the old ones. Read ξ η ξ x − c t = characteristic line — a straight track in the x t
On the space–time floor, x − c t = const t x c x + c t = const x , t u ξ η = 0
Common mistake "Minus sign means it moves left."Why it feels right: minus feels like backwards. Fix: f ( x − c t ) x − c t t x grow to keep it constant → the shape slides in the +x (right) direction. Minus = moves right.
The last unfamiliar mark on the parent page is in D'Alembert's formula:
2 c 1 ∫ x − c t x + c t g ( s ) d s .
Definition Definite integral∫ a b g ( s ) d s signed area trapped between the curve g a b s dummy variable — just a placeholder name for "the point we currently add up"; it never survives to the answer.
Intuition Why an integral appearsThe initial velocity g displacement (a height). Adding up velocity over an interval = total distance = area under the velocity curve. That is precisely what ∫ g d s [ x − c t , x + c t ] domain of dependence .
Recall Units check (why2 c 1 2 1 g ∫ g d s c length = a displacement. Question — what breaks if you drop the c
Function of one variable f of x
Partial derivative u sub x and u sub t
Function of two variables u of x t surface
Second partials u xx and u tt curvature and acceleration
Wave equation u tt equals c squared u xx
Operators partial t plus or minus c partial x
Characteristic coordinates xi and eta
Definite integral area under g
Every arrow says "you need the left box to understand the right box." Follow them and you can read the parent note line by line.
Can you draw a bump f ( x ) Yes — the slope is f ′ ( x )
What does the curly ∂ Another variable is being held frozen while you differentiate in the chosen one.
In words, what is u t The velocity of a fixed point on the string — height rise per unit time, with x
Read u tt = c 2 u xx Acceleration of a point equals c 2
Why may we factor ∂ tt − c 2 ∂ xx a 2 − b 2 Because u x t = u t x ∂ x ∂ t
Which way does f ( x − c t ) Rightward (+x); to keep x − c t t x
What does ∫ a b g ( s ) d s s The signed area under g a b s
Why 2 c 1 2 1 Dividing by the speed c
Parent topic 4.7.11 → Transport Equation — the first-order pieces the operator factors into.Method of Characteristics — where ξ , η Domain of Dependence and Influence — the meaning of the integral's interval.Classification of Second-Order PDEs — why u tt = c 2 u xx Separation of Variables — Wave Equation — the Fourier alternative on bounded strings.Heat Equation — contrast: no characteristics, infinite speed. 
