The value of the make variable VPATH specifies a list of directories that make should search.
Most often, the directories are expected to contain prerequisite files that are not in the current directory; however, make uses VPATH as a search list for both prerequisites and targets of rules.

VPATH用于指定一個目錄列表，以便于make的時候在其中搜索。
這些目錄中一般包含一些make必須的文件。

The VPATH is a list of directories to be searched for missing source files (actually for missing prerequisites: they don't have to be source files, but VPATH and vpath are best only used for source files).

The vpath directive has both a search path and a pattern.
Only missing prerequisites matching the pattern are searched using the associated path.
So vpath makes it possible, for example, to specify just a path to search for header (.h) files:

vpath %.h /usr/include

The % is the wildcard and matches anything, so that vpath directive solves the problem of the missing string.h in the example Makefile.

three forms

The vpath syntax is a little more complicated than VPATH and has three forms:

1. vpath pattern path 

This sets the search path (colon or blank separated) for the pattern.

2. vpath pattern 

This clears the path for the specified pattern.

3. vpath 

Clears all vpath settings

examples

vpath %.h /usr/include
vpath %.c ../src
INCLUDE_DIRECTORIES = ../include
vpath %.h $(INCLUDE_DIRECTORIES)

ffmpeg的主Makefile中也用到了vpath:

MAIN_MAKEFILE=1
include ffbuild/config.mak

vpath %.c    $(SRC_PATH)
vpath %.cpp  $(SRC_PATH)
vpath %.h    $(SRC_PATH)
vpath %.inc  $(SRC_PATH)
vpath %.m    $(SRC_PATH)
vpath %.S    $(SRC_PATH)
vpath %.asm  $(SRC_PATH)
vpath %.rc   $(SRC_PATH)
vpath %.v    $(SRC_PATH)
vpath %.texi $(SRC_PATH)
vpath %.cu   $(SRC_PATH)
vpath %.ptx  $(SRC_PATH)
vpath %/fate_config.sh.template $(SRC_PATH)

References:

https://www.cmcrossroads.com/article/basics-vpath-and-vpath
https://www.gnu.org/software/make/manual/html_node/General-Search.html